Tables of Bessel Transforms

Fri~ Oberhettinger
Professor of Mathematics, Oregon State University, Corvallis, Oregon, U.S.A.
AMS Subject Classifications (1970): 33 A 40,44 A 05,44 A 20
ISBN-13: 978-3-540-05997-4 e-ISBN-13: 978-3-642-65462-6 001: 10.1007/978-3-642-65462-6
This work is subject to copyright. All rights are reserved, whether the whole or part of the matenal is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1972. Library of Congress Cataloll Card Number 72-88727. Softcover reprint ofthe hardcover 1st edition 1972
For Joyce
forms involving Bessel (or related) functions as kernel. The
following types of inversion formulas have been singled out.
J k
I' . J k
II'. 1 c+ioo
k f (x) = 27fT J g (y) (xy) 2 [Iv (xy) + I_v(xy)]dy c-ioo
or also
k f(x) = rri J g (y) (xy) 2Iv (xy) dx
c-ioo
II".
0
0
J k
Preface
V.
V'.
VI.
VI' .
VII.
VII' •
g(y) J f(X)Kix(y)dx 0
f(x) -2 sinh (7TX) J -1 27T x g(y)y Kix(y)dy 0
g(y) 21-~[r(~~+~-~v)r(~~+~+~v)]-1
. J f (x) (xy) ~s (xy) dx o ~,v
f(x) l-~ -1 2 [r (~~+~-~v) r (~~+~+~v) ] •
• J g(y) (XY)~[S~,v(xy) -5 (xy)]dy 0 ~,v
g(y) ~ J f(x)\ [xy)~]dX 0 0
f(x) ~ f g(y) \ [(xy) lz]dy 0 0
with \ (z) o
this book.)
The transform VII is also known as the divisor transform.
Greek letters denote complex parameters within the given
range of validity while latin letters signify positive real
numbers. A possible extension to complex values will in general
require a minor effort. In a few cases the expression for g(y)
is given only for a part of the internal (0,00) for y. This
means that g(y) cannot be given in a simple form for the
VI Preface
involving Bessel functions as integrand (not necessarily of one
of the transform types I-VII) include the work by Y. L. Luke
(Integrals of Bessel functions, ~ew York, McGraw-Hill, 1962,
419 p.) and A. Erdelyi et. al. (Tables of Integral Transforms,
Vol. 2. New York, McGraw-Hill 1954, 451 p.). Compared to the
latter (pp. 1-174) the material displayed here represents a
considerable extension. Large parts of it do not seem to have
been available before.
Oregon State University
1. 2 Transfonns of Order Zero........................................ 6
1 .3 Transfonns of Order Unity....................................... 28
Transfonns of General Order
1.5 Exponential and Logarithmic Functions ........................... 45
1.6 Trigonometric and Inverse Trigonometric Functions ............... 49
1 .7 Orthogonal Polynomials.......................................... 64
1 .8 Miscellaneous Functions......................................... 68
1 . 9 Legendre Functions.............................................. 71
1.10 Bessel Functions of Argument x.................................. 80
1 . 11 Bessel Functions of Other Arguments............................. 96
1. 12 Modified Bessel Functions of Argument x ......................... 108
1.13 Modified Bessel Functions of Other Arguments .................... 117
1. 14 Functions Related to Bessel Functions ........................... 124
1. 15 Parabolic Cylinder Functions .................................... 129
1. 16 Whittaker Functions... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 136
1.17 Gauss I Hypergeometric Function.................................. 143
VIn Contents
as KerneL ................................................ 145
2.2 Transforms of Order Zero........................................ 148
Transforms of General Order
2.3 Elementary Functions............................................ 156
Chapter III. Integral Transforms with Neumann Functions as Kernel ..... 191
3. 1 General Formulas................................................ 192
3.2 Transforms of Order Zero........................................ 193
3.3 Elementary Functions............................................ 196
Chapter IV. Integral Transforms with Struve Functions as Kernel ....... 221
4.1 General Formulas ................................................ 221
4.3 Elementary Functions..................... . . . . . . . . . . . . . . . . . . . . . .. 226
Contents IX
Chapter VI. Transforms with Lommel Functions as Kernel ................ 262
Chapter VII. Divisor Transforms ....................................... 267
Appendix. List of Notations and Definitions .....................•..... 277
Chapter I. Hankel Transforms
(1)
(2)
The representation of a given real function f(t) of the real variable t by means of a double integral involving Bessel func tions of order v is known as Hankel's integral formula
f(x) = J Jv(tx)tdt J f(u)Jv(ut)udu o 0
Equivalent with this is the pair of inversion formulas
9 (y; v) J f(x) (xy)~J (xy)dx o v
f(x) l< J g(y;v) (xy) 2J V (xy) dy
o
which represent the Hankel transform of a given function f(x) and its inversion formula. The Hankel transform is self recip rocal and since
it is obvious that the Fourier sine transform 9 (y) I the Fourier cosine transform 9 (y) and the exponen~ial Fourier c transform ge(y) of a function f(x) are special cases of
(1) and (2)
(~TI)-~ J f(x)cos(xyldx g(y,-~l o
J f(xleixYdx o
2 I. Hankel Transforms
The two dimensional Fourier transform of a given function f(x,y) of two variables defined by
F(x,y) f f f(x' ,y')eixx'+iYY'dx'dy'
leads if f(x,y) only i.e.
is such that it depends on p
F(x,y) F (p) ~ k k
21fp-2 f p'2f(p')(pP')2JO(pp')dp' o
The integral occuring here is the Hankel transform of order zero k
of the function p2f(p). Similarly for the three dimensional Fourier transform of a function of three variables f(x,y,z) such that
F(x,y,z)
f f f f(x' ,y' ,x')eixx'+iyy'+izz'dx'dy'dz'
F(x,y,z) F(R) 41fR- l f R'f(R')sin(RR')dR' R'=O
The integral here represents the Fourier sine transform (or the Hankel transform of the order v=Y,) of the function Rf(R).
In this connection it should be pointed out that in Poisson's summation formulas in one, two or three dimensions
F (x) 1
n=-oo A e n
-i21fn~ a
-i21fn~ - i21frnt-a
I. Hankel Transfonns
F 3 (X,y,Z) L L L f 3 (x+na,y+mb,z+kc) n=-oo m=-oo k=-oo
\ \ \ -iTIn~ -i2~~~ -i2TIk~ l l l Anmke a 0 c
n=-OO m=-oo k=-oo
the coefficients
H (u,v) 2
H (u) 1
f f -00
H (u,v,w) 3
f J f f ( ) iux+ivy+iwzd d d 3 x,y,z e x y z
which are the Fourier transforms (in one, two or three dimen sions) of the functions f, f , f involved in the summation.
1 2 3
Erde1yi, A. et.a1., 1953: Higher transcendental functions, Vol. 2. McGraw-Hill, New York.
Erde1yi, A. et.a1., 1954: Tables of integral transforms, 2 vo1s. McGraw-Hill, New York.
3
Sneddon, I. N., 1951: Fourier transforms. Mc.Graw-Hill, New York.
Titchmarsh, E. C., 1937: Introduction to the theory of Fourier integrals. Oxford.
Watson, G. N., 1922: A treatlse on the theory of Bessel functions. Cambridge.
1.1 General Formulas 5
1.1 General Formulas
'" f (x) (xy) ~J v (xy) dx f (x) g(y;v) = J
0
0
a > a g(ya ;v)
~-v d m m+v-I<
1.3 xmf(x) ,m=O,1,2,··· y (yay) [y 2 g (y; v+m) ]
(_l)m ~+v (...£... m m-v-I<
1.4 xmf (x) ,m=O, 1,2, ••• Y ydy) [y 2 g (y;v-m)]
1.5 -1 2vx f(x) yg(y;v-1)+yg(y;v+1)
x -Il f (x) 21-Il[r(Il)]-ly~-V . 1.6
Rell>O,Re(v+1»Rell J V-Il+1< 2 2 11-1 T 2(y -T) g(T;v-ll)dT 0
x- Il f(x) 21-Il[r(Il)1-ly~+V . 1.7
00
o ,ReV-~2>Rell I -V-Il+1< 2 2 11-1 Rell > T 2 (T -y) g (T ; v+ll) d T
y
"
2.1 x-~ Y -~
x-~ k k x < 1 y 2J 0 (y) +~TIY 2 [J 1 (Y)Ha (Y) -J 0 (Y)K:t (Y) 1
2.2
0 x > 1
0 x < 1 -~ ~
Y [l-J 0 (Y) 1 +~TIY [J 0 (y)H1 (y) -J 1 (y)HO (y)]
2.3 x-~ x > 1
x-~(a2+x2)-~ y~ (l+e -ay)
2.5 [a2+x2)~+xl .
2.6 2 2 k -1 • [(a +x ) 2+xl
x~(a2-x2)-~ x < a -k y 2 sin (ay)
2.7 0 x > a
x -~ (a 2_X 2) -~x ~TIY~[Jo (~ay) 1 2
2.8 < a
00
f (x) (xy) \J 0 (xy) dx f (x) g(y) = f 0
x \(a2_x2)].l, x < a ].l ].l+1 -].l-k 2 a y 2r(].l+1)J (ay) ].l+1 2.9 0 x > a
Re ].l > -1
x\(x2-a2)-\ y
x-\(x2-a2)-\ x > a -\nY\Jo (\ay)YO (\ay)
2.12 x\(a~+x~)-l -2 k -a y 2kei o (ay)
2.13 x 5/z (a~+x~)-l y\kero(ay)
5/2 (a~-x~)-l x -\Y\[Ko(ay) - \nYO (ay) ]
2.14 Cauchy principal
value
2.15 x-\ (a 2 +x2)\_x -\ -2 _5/2 -ay e -aY_l) (a2+x2)~+x
y +2a y (aye +
x\(x~+2aZx2+b~)-\ k -k \ y 2r 0 [2 "y (a 2 - b 2) ].
2.16 Ko[2-\y(a2+b 2 )\] a > b .
0
x~(x4+2a2x2+b4)-~ y~JO[2-~Y(b2-a2)~] . 2.17
b > a -1< 2 1< . KO[2 "y(b+a2)2]
x\l+x4) -~ ~ -1< -~ . Y J (2 'y)K (2 y) v v
2.1B . [x2+ (l+x4)~]-V
2.21 - 3;2 -ax x (1-e )
;. -1 y 'sinh (ay )
~ 1< - L. 2 x-~e-ax2 ~(!.) y' e 8a I (L.)
2.22 a o Ba
-11k - ~ x 2Y~Ko[(2aY)~]Jo[2ay)~] 2.24 x e
-1 -ax~ 1 -~ a 2 2 a 2 2 2.25 x e 8" nay {[JJ.., (By)] + [YJ.., (8y)] }
1.2 Transforms of Order Zero 9
00 k f (x) g(y) = f f(x) (xy) 2JO(xy)dx
0
k k x2exp[-a(b2+x2) 2] ljj % k ay (a 2+y2)- [l+b(a2+y2) 2]
2.26 k . exp[-b(a2+y2) 2]
xljj(b 2+X 2)-ljj . yljj(a 2+y2)-ljj exp[-b(a 2+y2)ljj] 2.27 . k
· exp [-a(b 2+x2) 2]
2.28 x-ljjlog x - y-ljj log (2yy)
x-ljj(a 2+x 2)-ljj . k 2.29 k y2[ljjK~(ljjay) + log a IO(ljjay)KO(ljjay)]
· log[x+(a2+x2) 2]
x-ljj(a 2+x 2)-ljj . yljj K2 (ljjay) 0 2.30
(a 2+x2)ljj+x ·log [ J<] (a 2+x2) 2_X
xljj(z2-l)-ljj . 2.31 · log [z+ (z2-l) ljj] k 2aby 2 KO(ay) KO(by)
z = (2ab)-1(a2+b 2+y2)
xljj(a"+x,,)-ljj k -k -k 2.32 - ljj1TY 2 YO (2 2ay) KO (2 2ay)
x 2+(a"+x,,)ljj · log[ ] a 2
f(x) (XY)~JO(XY)dX f (x) g(y) = f 0
~ -2 -~ -1 -aKl (ay) 1 2.33 x log(1+a 2x ) 2y 2[y
2.34 ~ -1 2 ~ x'iog[ax +(l+a2x-) 1 - 3/2 -ay
y (l-e )
x-~e-ax2 - ~ 2 2.35 log x 2a-1y~e 4a[log (!a) - ~ Ei (~) 1
x-~ sin (ax) y~(a2_y2)-~ y < a 2.36
0 Y > a
2.37 y~(y2_a2)-~ y > a
- 3h sin (ax) ~ ~ x 27fY Y < a
2.38 ~ -1 Y arcr:;in (ay ) y > a
-% -~ a a 2 1)~1 x [I-cos (ax) 1 y 21og[_ + (- - y < a Y y2
2.39
2.40 x-~(b2+x2)-1sin(ax) y~b-1sinh(ba) KO(by) y > a
2.41 x ~ -1 (b 2+X 2) sin (ax) ~7fY~ e -ab
10 (by) y < a
'" f (x) g(y) = f f(x) (xy JO(xy)dx 0
2.42 X-\(b2+X2)-lcos(ax) \b-1ny\e-abI O(bY) y < a
x\(b2+x2)-lcos(ax) l< 2.43 Y 'cosh (ba) KO(by) y > a
x\sin(ax2) -1 l< ~ 2.44 \ a y 'cos ( ) 4a
2.45 x\cos (ax 2) -1 l< \a y' sin
y2 (4a)
4a 2
4a 2
x-Is in (ax\) -\ a 2 a 2 a 2 2.4S \nay J\(Sy) [J\(Sy) - Y\ (Sy)]
x -lcos (ax\) 1 -l< a 2 a 2 a 2 snay 2{y\(SY) [Y\(Sy)-J\(Sy)] -
2.49 a 2 a 2 a 2 - J\ (Sy) [J\ (Sy) + Yk(B)]} 4 y
-1 -ax\ l< -lar a 2 a 2 2.50 x e sin (ax') \ay \(4y) K\(4y)
-1 -ax\ l< a 2 a 2
2.51 x e cos (ax\) \ay- 2 1 k(4) K\(4y) -4 y
12 I. Hankel Transfonns
X-~cOS(aX)lOg(bX) -~lTy~(a 2_y 2)-~ Y < a
2.52 -y~(y2-a2)-~[Y~log(~Y)+log(y2_a2)1
y > a
x~(b2+x2)-~sin[a(b2+x2)~1 y~(a2-y2)-~cOS[b(a2-y2)~1 y < a 2.53
0 Y > a
x~(b2+x2)-~cos[a(b2+x2)~1 -y~(a2-y2)-~sin[b(a2-y2)~1 y < a 2.54
y~(y2-a2)-~exp[-b(y2-a2)~1 y > a
x~(a2-x2)-~cos[b(a2-x2)~1 y~(b2+y2)-~sin[a(b2+y2)~1
2.55 x < a
0 x > a
0 x < a 0 y a
x > a
x~sin[b(a2-x2)~1 ab(b2+y2)-1{a-l(b2+y2)-~sin[a(b2+y2)~1_
2.57 x < a - cos[a(b2+y2)~1} 0 x > a
x-~e-axsinh(bx) (aby)~(uv)-l(u-v)~(u+v)-~ 2.58
(b-a)21~, v=[y2+(b+a)21~ a~ b u= [y2+ - - ---------
'" f(x) (xy)~JO(xy)dx f(x) g (y ( = b
x-~e-axcoSh (bx) ~ -1 J.: -l< (aby) (uv) (u+v) 2(U-V) 2
2.59 J.: l< a ;:,b u = [y 2+ (b-a) 2] ., v = [y 2+ (b-a) 2] •
- ./z -ax ~~log[2a + 4 2 l<
2.60 x e sinh (ax) (l+~) 2] Y
Y
k -ax -1 -1 -\1- ~ ~ 2.61 }< 2e sinh(ax ) 2ay 1 [2 (ay) ] K1 [2 (ay) 1
x\> 2 < 1 -~ n(1-2x ) x y J 2n+1 (y)
2.62 0 x > 1
n = 0,1,2, •••
xSlzp n (1-2x 2) x < 1 -~ -1 y (2n+1) [(n+1)J~n+2(y)-nJ~n(Y)1
2.63 0 x > 1
x~ (a 2+X 2) -lon-~
p [a (a 2+X 2) -~l 1 n-k e -ay 2.64 n n! y 2
n = 0,1,2, ...
Ox> 1
n=0,1,2,'"
x < 1
Ox> 1
n = 0,1,2,'"
x~exp (a 2X 2)
2.71 • Erfc (ax)
I. Hankel Transfol1JlS
g(y) = J f(x) (XY)~JO(XY)dX o
[ ( 2 2 -~ [( 2 2 ~ .P2n+l a a +y) ]J2n+ 3/ 2 a +y ) ]
-1 -2n-l -2n-2 2n+~ • (n!) 2 a y
-~ -1 Y Erfc\~a )
1.2 Transforms of Order Zero lS
'" k f (x) g(y) = f f (x) (xy) 2J 0 (xY) dx
0
2.74 ~ - S (a2x 2) y-~C(~y2a-2)
2.75 ~ - C (a2x 2) -~ y S (~y2a -2)
2.76 x~Ei (-ax 2) _312 -1
-2y [1-exp(-~y2a)]
2.77 k . x2exp(ax2)E~(-ax2) -1 ~ 2 -1 2 -1 (2a) y exp(~y a )Ei(-~y a )
k -2k-1 -2-2k [r(k+1)]-1y2k+~ 2.78 x 2exp (-ax 2) 2 a
• Lk (ax2) 2 -2 . exp (-~y a )
Re k > - 3/ ..
0 y > a
sin (4a)
2y [1-cos(4a)]
00
f (x) (xy) ~J 0 (xy) dx fIx) g(y) = f 0
-k -k r..:£ 2 2.82 x 'ci (ax 2 ) y 2[log( ) + Ci (~a) ] 4a
X~(1+x2)-V-1 . 2.83 . P [(1-x 2 ) (1+x2) -1] [2 Vr(v+1)]-2 2v+~K ( )
v y 0 y
-~ Y W (ay) 2.84 v-Jo ~,v
Re ~<1, -J,o < Re v < J,o [W (ayeiTT ) ~,v
- W (aye -iTT) ] ~,v
2.85 p-~ [(1+x 2 a -2) 10] v-Jo
-1;o<Re v < 1;0
k ~ -2 k ei1[~ r(Jo+v-~) - 3/2 x'p [(1+x2a) 2] . Y v-Jo r(1+2v )
2.86 q~-Jo[ (1+x2a-2)10] . W (ay) M (ay)
~,v -~,v
f(x) g(y) k
3/2 1T-1 r (3/, ) ~ (a2_y2)-~ J 0 (ax) 2 (¥o) .
r(l:d a
• K.{[~-~(l-y a )2]2 } y < a
2 3/21T -1 r (314) (y2-a2 ) -~ fT!i) .
• K {[~_~(1_a2y-2)~]~} y > a
-~ 2 (ITa) -ly~ K.(~) x J O (ax) y < a
2.88 2IT -ly -~ K(~) y > a
y
-1 2~1T-1 r(la) -2 ~ x J O (ax) r ( 3/ .. )
(1-y 2a )
2~ IT- 1 r(la) au ~_~ (1-a 2y -2)~]~} r (~A)
Y > a
~ log (~) Y > a y y
2.91 -~ -bx x e JO(ax)
-1 k 2 2-k 2 IT Y 2 [ (a+y) + b ] 2
• K{2(ay)~[(a+y)2+ b2]-~}
DO k f (xl g(y) = f f (x) (xy) 2J 0 (xy) dx
0
_ 812 J 1 (ax) 21T-1y~ £(l.) 2.92 x y < a
a
-1 3/2 a (1_a2y-2)£(~)] > a 2 (1Ta) y [K(y) - y
X-~[Jo (~ax)] 2 -2 -k 2 2 -2 k k
2.93 41T Y "It {[~-~(1-a y ) 2] 2} Y > a
2.94 k x 2JO (ax) J v (ax) -1 -k 2 2-~ 21T y 2(4a -y) cos [varccos (h) ]
Re v > -1 y < 2a
0 y > 2a
k 2.95 x 2J O(ax)Yo (ax) 0 y < 2a
-1 -k 2 2-k -21T Y 2 (y -4a) 2 y > 2a
2.96 k _(1Tab)-1Y~(Z2_1)-~ x 2JO (ax) YO (bx) y < a-b
1
-(1Tab)-1y~(Z~-1)-~ y > a+b
Zi = ± a 2+b 2_;( 2
2ab 2
2.97 k
X 2J O (bx)YO (ax) (1Tab) -1y~ (z ~-1)-~ y < a-b
a > b 0 a-b < y < a+b
_(1Tab)-1Y~(Z;-1)-~ y > a+b
Z = ± a 2+b 2_;( 2
1 2ab 2
f(x)
2.100
Re II > -1
o
o
-1 -k 2 2-k 211 Y 2 ( 4 a _Y ) 2 •
• COS[llarccos(1-~y2a-2)]
2.102
2.103 Re II > -1
y < 2a
• [~- 1+ ~(1_4a2)~]-1l y > 2a 2a2 2a2 y2
o
_('JIab)-1Y~(Z~-1)-~Sin('JIIl) •
2 2 4 2 t- • {[:L.:.- -1+ :L.:.-(1- ~) '2] II
2a 2 2a 2 y2
_ [~_ 1+~ (1_4a2)~]-Il} 2a 2 2a 2 y2
1 .2 Transfonns of Order Zero 21
f (x) g(y) k
X~J (bx)Y (ax) II J.l
2.105 Re II > -1 y < a-b
a > b
a-b < y < a+b
-1 k 2 -k k -ll - (1Iab) y 2COS (1IJ.l) (z -1) 2[Z + (z2-1) 2]
2 2 2
y > a+b
x~J (ax)Y (bx) J.l J.l
-1 k 2 -k 2 k J.l - (1Iab) y 2 (z -1) 2 [z + (z -1) 2]
1 1 1
the same values as before
2.107 k -1 2 3 k 2 2 -k 2 2 k -k = 2'11 r ('4)y2(a +y) '[y+(a +y )'] •
k 2 2 k -k . K{ (2y) 2[ (a +y ) 2+y ] 2}
. K{ [ (a2+:i2)~-a ~
• K { (2y)~[(a2+y2)~+y]-~ }
2.110 X~KO (ax) y~(a2+y2)-1
2.111 x~Jo(aX)Ko(bX) y~(a4+b4+y4-2a2y2+2a2b2+2b2y2)-~
2.112 a/2
x J 1 (ax (Ko (bx) ~ -~ 2ay (a2+b2_y2) [a2+b 2+y2)2_4a 2y2]
2.113 X~1o(aX)Ko(bX) y~(a4+b4+y4_2a2b2+2a2y2+2b2y2)-~
b > a
2.114 a/2
x 10 (ax) Kl (bx) ~ -ah 2y b (b2+y2-a2([(a2+b 2+y2) 2-4a2b 2]
b > a
1 .2 Transfonns of Order Zero 23
f(x) g(y) = j f (x) (xy) ~J 0 (xy) dx 0
x~I 1 (ax) K 1 (bx) (2ab) -1y~ {(a2+b2+y2) . 2.115 b > a 2 _~
• [(a 2+b 2+y2) -4a 2b 2] _ 1}
x -~o (ax) KO (ax) 1I-1a-2y~ [(4a2+y2) \...Y]K{ 2a [(4a2+y2. )~+y]-~r
• K{(2y) ~[(4a2+y2 )~+y] -~ }= 2.116
-1 - ah ~ ~ - ~ =411 Y [(4a 2+y2} -2aJlC{y[ (4a 2+y2} +2a]
• K{2a ~[ (4a 2.+y 2.fz +2a] - ~ }
x -~ [KO (~ax) ] 2. 2y~a -2 [(a2+y2) ~-Y]K2.{ (2y) ~[(a 2+y2.) ~+y] -~}
2.117 =4y - ak [(a 2.+y2.) ~-a]a.a {y [ (a 2+y2.) ~+a] -~}
2.118 x ltyo (ax) KO (ax) -1 ~ -~ ~ (1~ ~] -211 Y (y4+4a~) 1og[ +
2a2. 4a 4
x~Ko(aX)Ko(bX) ~(ab)-1y~(z2.-1)-~109[Z+(Z2_1)~] 2.119 -1 Z = (2ab) (a2+b2.+y2)
2.120 x ~Kll (ax) III (bx) ~(ab)-1y~(z2-1)-~[z+(z2.-1)~]-1l
a > b, Re II >-1 z = (2ab)-1(a2.+b 2+y2.)
X~Kll(ax)Kll(bX) ~1I(ab)-1CSC(1Ill)y~(z2.-1)-~ •
0
I< _3/2 -1 -I< 1 2 -2 _312. a-1y) x 2exp(a2x 2) • ~1f a y 2exp (gy a )Ertc(2
2.122 . K (a2x 2) 0
-~ -1 -1 -~ -1 -2 a ~ I< 2.123 x I k(ax )Kk(ax ) aye (y) cos [2 (ay) 2]
-4 4
-I< -1 -1 I<
-~ -1 -2 (ay) 2 sin[2(ay)~] 2.124 x ·1k (ax (Kk (ax l aye 4 4
-I< -1 2 ~ x 2 [Kk (ax l] 1f(2al-~y-1e-2(aYl .
2.125 4 I<
2.126 I< -ax 2 x·e 1O(ax2l (21fay)-~exp(- ~ 8al
I< 2 -1 I< L L 2.127 x 2[JO(ax 2 )] -~a y 2J ( l YO (16a l o 16a
2 2.128 I< -1 I< L x 2J O(ax2)YO(ax 2) -~a y 2 [J ( l] o 16a
I< -1 I< L L 2.129 x·1O(ax2lKO(ax2l ~a y 2KO (l6al 10 (l6a l
X~J (ax~l K (ax~) _3~ -~a2y -1 2.130 ~y e o 0
1 .2 Transfonns of Order Zero
f(x)
2.134 Re ~ > -~
2.135 - ~ < Re ~ < ~
-2~Y-~K (2ay~) • 2~
2S
-1 -~ 2 2-1 - ~~ y KO(~a y )
co I< f (x) g(y) = f f (x) (xy) 2J 0 (xy) dx
0
X-~J ].I(aX~)K].I(aX~) -~ 2 -1 2 -1 ~y I~].I(~a y )K~].I(~a y )
2.137 Re ].I > -1
2.138 X-~Y (ax~)K (ax~) -~-~seC(~U].l)K~].I(~a2y-1) . ].I ].I
-1 < Re ].I < 1 -1 2 -1 2 -1 • [u K~].I(~a y )+sin(~U].l)I~].I(~a y )]
.U _I< ~4 ~
H(l) (k 2 -1)H(2) (k 2 -1) x 2K (ae x) . i6u2sec(~u].I) ].I ~].I 4a y ~].I 4a y
.u -~-
- 1 < Re ].I < 1
-1 I< -1 -1 4u y2(a+y) K[la-yl(a+y) ]
2.141 X-~[Io(ax)-Lo(ax)] 2u-ly~(a2+y2)-~K[a(a2+y2)-~]
2.142 x-10 (a~x~)o (a~x~) 2-3hu(~)~p~V+~[(1+4y2a-2)~] . v -v-l a -~
· p-~v-~[(I+~y2a-2)~] -~
· -~ 2 -2 ~ q-~v- 3/4 [(l+~a y ) ]
-k -2 I< · q 4 [(1+~a2y )2] ~v-~
1 . 2 Transfonns of Order Zero
f(x)
-\i 2.145 x Dv (-x) Dv- 1 (x)
_ 3/2 x Wk (ax)M k (ax)
I~ -,~
g(y) k
= I f(x) (xy) 2JO(xy)dx o
Y-\i[D (y)D l(Y)+\iD (y)D l(-Y) + v v- v v-
y\i[\iDv (y)Dv- 1 (-y)+\iDv (-y)Dv- 1 (y) -
- Dv (y)Dv_1 (y)]
= e-iTI~ r(l+2H) ay-\i. r(\i-k+~)
• p-~ [(1+a2y-2)\i]q~ [(1+a 2y-2)\i] k-\i -k-\i
_3k \i x Wk (ax)W k (ax) \iTIcos(TI~)Y
I ~ - I ~
2.147 -\i < Re \.I < \i Pk [(1+y2a -2) \i] p -k d (l+y2a -2) \i] ~-\i ~-'2
= ei2TI~cos(TI~) [r(\i-\.I+k)r(\i-~-k)]-l
.ay-\iq=~_\i[(1+a2y-2)\i]q~~\i[(1+a2y-2)\i]
27
00 1,
f (x) g(y) = J f (x) (xy) '.J 1 txy) dx 0
x-~ x < a y-~[l-Jo(aY)l 3.1
0 x > a
x-~ x > a
0 x > a
x-~(x2-a2)-~x > a
x-~e-ax2 k 2 3.7 Y - 2 [l-exp (_:L:.) 1
4a
3/2 -ax 2 -2 3/2 2 3.8 x e l;,a y exp (_:L:.)
4a
3.9 -1: k
x 2exp[-a(b 2+x2) 2] -~ -ba 2 -k k y {e -a(a +y2) 2exp [-b(a 2+y2) 2J}
1 .3 Transfonns of Order Uni ty 29
'" f (x) (xy) ~J (xy) dx f(x) g(y) = f
0 1
y [e -e ]
a _ 312 -~
1 1
_ 7/2 a x -1 3, k k
3.12 x e 2a y 2J [(2ay) 2]K [(2ay)2] 2 2
-k -y-~log(~yy) 3.13 x 210g x
3.14 x -~log (a 2+X 2) 2Y-~[Ko(ay) + log a]
3.15 -k 2 k
x 'log [ax+ (l+a x 2) "] -~ y L y 10 (2a) KO (2a)
3.16 x -~log (l+x~) -k 4y 2keroy
3.17 x -~sin (ax 2) -k 2
y 2sin (l.::.) 4a
y2 (Ba)
3.19 x -~sin 2 (ax 2) y-~cos 'L (Ba)
x-~(a2+x2)-~ . a-1y-~{sin(ab)-sin[a(b2-y2)~1}y b
x-~(a2+x2)-~ • a-1y-~{cOS(ab)-Cos[a(b2-y2)~1}y b • cos
x-~(a2-x2)-~COs[b(a2-x2)~1 a-1Y-~{COS(ab)-Cos[a(b2+y2)~1}
3.22 x < a
0 x > a
0 x < a 0 y b
x > a
3.24 x-~(x2-a2)-~ • a-1y-~sin[a(y2-b2)~1 y > b
• cos[b(x2-a2)~1, x > a
-~ -ax2 -~ -1 x e sin(bx 2) y exp[-~ay2(a2+b2) 1 . 3.25
• sin[~y2(a2+b2)-11
'" f(x) (xy)~J (xy)dx f (x) g(y) = f 0 1
x-~e-ax2sinh(bX2) y-~exp[_~ay2(a2_b2)-l) . 3.26 -1 a > b • sinh[~y2 (a 2-b 2) )
3.27 x-~arctan x 2 - 2Y~keiOY
x- 312p n (1-2x 2) x < 1 (2n+1)-ly-~[(n+1)J2n+2(y)-nJ2n(Y») 3.28
0 x > 1
-~ y Y > a
a a y < a 3.32
-1 ~ a 21T y (y) Y > a
_5/2 '/2 10g(~) ) x [J 0 (ax) -1) -~y [1+2 y < a 3.33
-~y~a2 y > a
co k
f (x) g(y) = J f (x) (xy) 2J (xy) dx 0 1
-k -1 -k 2 -2 3.34 x 2yO (ax) -TI y 21og(1-y a ) y < a
3.35 x~keiox -~y -~arctan (y2)
3.36 -k x 2kerOx -k
!,jy 21og(1+y2)
-k la-bl x 2J O(ax)JO(bx) 0 y <
-1 -k -1 TI Y 2arccos[(2ab) (a 2+b 2_y2)]
3.37 la-bl < y < a+b
y -~ y > a+b
a
3.38 X~J [(ax)~]K [ (ax)~] k _3/2 -2y 2Y e
1 1
2 2
-k y-~log[:L + (l+~)~] 3.40 x 210 (ax) KO (ax) 2a 4a 2
X-~l -~ -1 [:L + 2 k
(ax)K (ax) ~y ]J {l- (l+l.:-) 2]} ]J ]J 2a 4a 2
3.41 Re ]J > -1
ex> k
f(x) g(y) = f f (x) (xy) 2JV (xy) dx 0
1, 0 1 2l>;; r (!j,~+l>;;v) -1 (v-l>;;)Jv(Y) . < x < r (Vl>;;v) y +
4.1 0, x > 1 . S-l>;;,v-l (y)-Jv- 1 (y)Sl>;;,v (y) Re v > - 3~
x-l>;; y-l>;; 4.2
Re v > -1
xl>;;-v I-v v-~ -l>;;
I 0 < X < 1 2 Y - y J v- 1 (y) 4.3 r(v)
0 , x > 1
xv-l>;; , 0 < x < 1 2v- 11Tl>;; r (l>;;+v) . 4.4 0 , x > 1 l>;;-v [Jv (y)llv_l(y)-llv (y)Jv_1 (y)] y
Re v > - l>;;
xv+l>;; , 0 < x < 1 y-l>;; J (y) v+l
4.5 0 , x > 1
Re v > -1
xjl 2jl+~-jl-l r (l>;;jl+l>;;V+3~ 4.6 r (l>;;v-l>;;jl+!;;)
- Re V_3~< Re jl < 0
f(x) (XY)~Jv(XY)dx f (x) g(y) = f 0
x].l 0 1 -].1-1 -, < x < Y [(v+].I-~)yJv(Y)S].I_~,V_1(Y)
4.7 0 , x > 1 -yJv_1(Y)S].I+~,v(Y) +
Re (].I+v) > - %
r (liiv - ~].I + J.i j
x-~ -1 k 4.8 (a+x) lTCSC (lTv) y 2 [.:Iv (ay) - J'v(ay)]
Re V > -1
xV-~ -1 ~lTav SeC(1TV)y~~(ay) - Y- v (ay) ] (a+x) 4.9
- ~ < Re V < 312
x].l-~ -1 (2a) ].Iy~ [r(~+~ll+~V) s_].I,v(ay ) (a+x) r(lii-lii].l+liivj -
4.10 Re(].I+v) > -1
- 2 r(l+~].I+~v) S-].I-1,V (ay) ]
Re ].I < 3/2 r(~v-lii].l)
x-~ (a )-1 k -1 +x ~y 2a IT{sec (~lTv) Iv (ay) +CSC (lTv) [Jv (iay)- 4.11
Re v > -1 - \I_v(iay)]}
Re v > -2 - 2 COS(~lTV)Iv(ay)]
1.4 Algebraic Functions and Powers with Arbitrary Index 35
00
f (x) (xy) ~J (xy) dx f{x) g{y) = f 0 v
x v+~ (a2+x2)-1 aVy~ Kv{ay) 4.13
-1 Re v < 3/2
xV-~{a2+x2)-1 ~1Tav-1 SeC{1TV)Y~[IV{aY)-L_v{aY)l 4.14
-~ < Re v < 512
Re v >- 512
lJ-312 -1 x (a2+x2) lJ-2 ~ ~1Ta csc[~1T{v+lJ)ly IV{ay) - 4.16 -Re v < Re lJ < 7/2 11-3 r{~v+~lJ-1) y~alJ-2 (2+v-lJ) (2-V-lJ) . -2
r{~v-~lJ+2)
.1T . J.'2lJ sl_lJ,V(iay ) e
x-~(a2+x2)-~ k K~V{~ay) y 2I~V (~ay)
4.17 Re v > -1
-1 < Re v < ~
4.19 x~-v (a2+x2)-~ k k-v (~1T) 2a 2 [IV_~ (ay) - "'V-~ (ay) 1
Re v > -~
'" k f(x) g (y) = f f (x) (xy) 2J v (xy) dx
0
x - v- >'(a 2+X 2) -lr v 2-VTf>'a- 2V >,+V IV (>,ay) KV(>,ay) r (li+v) y 4.20
Re v > ->,
4.21 r(Vv)
Re v > ->,
x v+>, (a 2+X2) -v- 3/2 2- V- 1Tf\,v+>'e-ay
4.22 ar (%+v)
Re v > -1
xv+ ~a2+x2)-].I 1-].1 v-].I+l ].1->, 2 a y.
KV-].I+l (ay) r(].I)
4.23 Re v > -1
Re (2].1-v) > >,
x>,-v (a 2+x 2)-].I a-].I-v+l ].1->, [2-].1 r (1-].1) I v+lJ - 1 (ay) y (I-v)
21- v -i2!.(v-lJ+l) 4.24 Re (v+2].1) > >, - r(v)
e 2 s_lJ+v,_lJ_v+l(iay )
x"-~(a2+x2) -lJ -1 2-v- 1 a,,+v-2-2].1y>'+V r(>'''+>,v)r(l+lJ->'''->'V). r (1+lJ)-r (1+v)
-Rev<Re,,<2RelJ+ 7h 4.25 'IF2(>,,,+>,v;>,,,+>,v-lJ,l+v;~a2y2) +
,,-2lJ-3 2lJ-,,+5 2 r (>,H>,v-lJ-1) + 2 y r(2+].I+>'v->,")
·IF2(I+lJ;2+lJ+>'V->,,,,2+].I->,v->''';~a2y2)
1.4 Algebraic Functions and Powers with Arbitrary Index 37
4.26
4.27
4.28
4.29
4.30
f(x)
g(y) = f f(x) (XY)~JV(XY)dX o
v+l ~ ~lTa sec(lTv)y . . [Jv (ay) sin (lTv) - Yv (ay) cos (lTv)
- H_ v (ay)] + 2vlT-~r(~+v)y-v-~
v ~ ~lTa sec (lTv) y 2 .
-
+ £v(ay)}
- H_ v (ay) sec (lTv) ]
f (x) (xy) ~Jv (xy) dx f(x) g(y) = J 0
_ 3 2 -1 -krra].l-2 y~y (ay)-x].l (x 2-a2) 2 v
4.32 Principal value _2].l-la].l-2 r(~v+~1!) y~Sl (ay) Re v < Re ].l < 7/2 r (I+~v-~].l) -].l,V
x -~(a 1_x 2 ) -\O<x<a
4.33 0 ,x > a ~'IIy~[J ~v (~ay) ] 2
Re v > -1
x~-v (a 2_X 2) -\ 0 <x <a (~'IIa)~ a -v H (ay) 4.34 v-~
0 ,x > a
x V+~(a 2_X 2) -v-~, 0 <x <a -~ v ('IIY) (~) r(\-v)sin(ay)
4.35 0 ,x > a
-~ < Re v < ~
xV+~(a2_x2)-v_3~o<x<a -~ -v-l -1 v+k 'II 2 a r(-~-v)y 2cos(ay)
4.36 0 , x>a
-1 < Re v < - ~
x~-V(a2-x2)].l,O<x<a 21 - va].l-v+l -U-~ SV+].l,].l-v+l (ay) r(v) y
4.37 0 ,x > a
00
f (x) (xy) ~J (xy) dx f (x) g(y) = f 0
v
xV+~(a2-x2)~,0<x<a ~ v+~+l -~_k 2 f (~+1) a y 2 J (ay) v+~+l
4.38 0 ,x > a
Re v>-l, Re ~>-1
x~-!o (a 2_X 2/, O<x<a 2-v-1 2A+~+v+1 f~v+1) B(A+1,~~+~v+!o) .
4.39 0 , x > a
Re A>-l,Re(~+v»-l .yv+~lF2(~+~~+~v;v+1,3~+3k~+~V+A;-~a2y2)
0 , 0 < x < a -!o1TY~ J~v(~ay) 4.40 y~v (~ay)
x-~(x2-a2)-~
0 , 0 < x < a
4.41 X-V-~(x2_a2)-V-~,x>a k -v-1 -2v v+~ -1T 2 2 a f(~-v)y Jv(~aY)Yv(~ay)
-~ < Re v < ~
0 , 0 < x < a
4.42 x!o-V(x 2_a 2)V-!o,x > a -k -v -v-k 1T 22 f(!o+v)y 2 cos (ay)
-10 < Re v < 10
0 , 0 < x < a
4.43 k-v v- 3Jz x 2 (x 2-a 2) ,x>a 1T-!o2- v- 1a-1 k-v f (v-~) y 2 sin (ay)
~ < Re v < %
'" k f(x) 9 (y) = J f (x) (xy) 2J v (xy) dx a
a , a < x < a
4.44 xV-l:!(x 2_a 2)V-l:!, x>a 1Tl:!2 v -2 r(l:!+v) a 2Vy l:!-v
-l:! < Re v < l:! . [J v (l:!ay)Y_v (l:!ay) +'Yv(l:!ay)J_v(l:!ay)]
a , a < x < a
4.45 xl:!-V(x 2_a 2»).I, x > a 2).1al +).I-V r(l+).I)y-).I-l:! J l(ay) V-).I-
Re ).1>-1, Re(v-2).1»l:!
a , a < x < a 2).1a1+).I+v r(l+).I)y-).I-l:!
4.46 xl:!+V(x 2_a 2»).I , x > a . [sin(1T).I)Y).I+v+1(ay)-cos(1T).I)J).I+v+1(ay )]
Re ).1>-1, Re(v+2).1)<-l:!
a , a < x < a y-2A-).I-1 2).1+2A r (Hl:!+l:!v+l:!).I) r (\v-\).I+\-A)
4.47 x).l-l:!(x 2_a 2)A, x > a ·lF2(-A;l:!V+l:!-l:!).I-A,l:!-l:!v-l:!).I-A; -l,;a 2y2)
Re ).1>-1, Re ().I+2A) <l:!
+ 2-v-1a2A+V+).I+1
r(l+v) B(l+A,-A-l:!V-l:!).I-l:!)
.yV1F2(l:!+l:!v+l:!).I;3h+l:!v+l:!).I+A;v+1;-l,;a2y2)
4.48 p_ 3h -).1-1 x (4a 4 + x 4 ) Watson, 1944, p. 435
Bessel functions
00 k
f (x) g(y) = J f (x) (xy) 2JV (xy) dx 0
Xv+5k(4a'+x')-V-~ 1I~2l-3va2-2v v+k 4.49 r(~+v)
y 2JV_l(ay)KV_l(ay) Re v > 1/6
xV+~(4a'+x')-v-~ ~ -3v -2v v+k 11 2 a
4.50 r(~+v) y 2Jv (ay)Kv (ay)
Re v > - ~
4.51 , O<x<l Math. Soc., 38, 177-180.
0 , x > 1
xV+~(x'±2a2x2+b')-~ . (b2"+a2) -V2Vy~
[b 2+X 2+ K [~b2±~a2)~Y1J [(~b2±~a2) v v
4.52 +(x'±2a2x2+b')~1-2v
O<a - ~
x-~(a2+x2)-~ . 4.53 [ (x2+a 2) ~+xl v-I -k v- 3/2
211 2a sinh(~aY)Kv_~(~ay)
-1 < Re v < 5/2
4.56
o
h h-v -hay 1T 2a 2 e 2 Iv_~ (~ay)
-~ -ay y e
4.59
4.60
4.61
f(x)
aVB(~+~V+~,~+~v-~) yV+~ r (1+v)
• IF1(~+~V-~; v+l; -iay)
IFI (~+~v-~; v+l; iay) =
= a-l[r(l+v)l-IB(~+~V+~, ~+~v-~)y-~
1< v-312 -n 2a [sin(~ay)Jv+~(~ay) +
+ cos(~aY)Yv+~(~aY)l
- sin(~aY)Yv_~(~aY)l
o
1.5 Exponential and Logarithmic Functions 45
1.5 Exponential and Logarithmic Functions
00 k f(x) g(y) = f f (x) (xy) 2J v (xy) dx
0
x -~e -ax k+v I I -k I k-V y2 (a. +y) 2[0.+(0. +yl) 2] 5.1
Re v > -1, Re a>O
- 3/1 -ax x e v-1y~+v[a+(al+yl)~]-v
5.2 Re v > 0, Re a > 0
xm+~e-ax ( I' m+1 ~+v dm+1 {(al+yl)-~ - I Y
dam+1 5.3 Re v>-m-2, Re 0.>0
m=0,1,2,··· . [a+(al+yl)~]-V}
x v+~e -ax 2 v+ 11[ -~r (v+ % ) ay~+v (a I+y I) -v- '0 5.4
Re v>-l, Re 0.>0
xV-~e-ax v -k v+k I I -V-k 2 1[ 2r (~+v)y 2(a +y) 2
5.5 Re v>-~, Re 0.>0
xj1-~e-ax y~(al+yl)-~j1-~r(V+j1+1)p-v[a(al+yl)-~] 5.6
Re(j1+v»-1,Re a>O j1
x-~e -ax l
5.7 ~(1[ Z. ) ~ ( yl
I~v ( L. ) exp - 80.) Re v>-1, Re 0.>0 a 80.
0 v
x\e -ax2 1 \ - 3/2 3/2 2 81T a y exp (-~)
5.S Sa Re v>-2, Re a>O L y2
[I\v_\ (Sa) - I\v+\ (Sa) ]
x\)+~e-O'.x2 (2a)-v-l yV+\ (- y2
exp 4a) 5.9 Re v>-l, Re c.>0
v- 3f2 -ax 2 2 v - 1 \-v y(v, L x e y 4a) 5.10
Re v>O, Re a>O
v+!.:: ±iax2 (2a)-v-l v+\ [+. (v+l ~)] x 2 e y exp _l. 2 1T - 5.11
-l<Re v<!z, a>O
x2n+v+\ e-~x2 22n+v+l n!yv+\ e-y2 LV( 2
) 5.12 n y
Re v>-1-2n,n=0,1,2,'
11-1: -ax 2 r(\+\v+\~) -\~ -\ L x 2 e r(l+v) a y exp (- Sa) 5.13 Re (~+v) >-l,Re a>O 2
M (~) \~,\v 4a
_3/2 - ::: ), x 2y\J [(2ay) \] 5.14 x e , Re a.>0 Kv [(2ay) 2] v
1.5 Exponential and Logarithmic Functions 47
f (x) g(y) 00
a _3/2 -X-/3x
x e 5.15
k k k k 2y 2JV {(2a) 2 [( S2+y2) 2-8] 2} •
Re a>O, Re 8>0 • K {(2a)!.z[(s2+y2)!.z+S]!.z} v
-1 -(ax)!.z x e
5.16 Re v>-!.z, Re a>O
, x<l (2 ' )-V-l !.z+V[ (2' )' (2' )] La y UV+1 La,y -LU v+2 La,y
5.17 o , x>l
Re V>-!.z
V+k 2 2 k X 2exp [-a(b +x ) 2] 5.18
Re v>-l, Re a>O
--~~----------------~--------------------------------------
00 h
f (x) g(y) = J f (x) (xy) 2JV (xy) dx 0
X~-V(b2+X2)-~ . v+h 2 h -v 2 -h Y 2 [a+ (a 2+y ) 2) (a 2+y ) 2 . '[(x2+b2)~_b)V h . exp [-b (a 2+y2) 2)
5.21 h 'exp [-a (x 2+b 2 ) 2)
Re v>-1, Re a>O
xT-~(b2+X2)-~ r(l-,+~V+~T) -~ h Y M, h {b[(a 2+y2)2_a)} .
br(l+v) ~'T, zV
[(b2+X2)~+b)-T . h
5.22 exp[-a(b2+x2) 2) -~T, 2'V
Re(V+T) > -1
Re a > 0
1.6 TTigonometric and Inverse Trigonometric Functions
00
f(x) (XY)~Jv(XY)dX f(x) g(y) = J 0
x -~sin (ax) v+~ 2 2 _k 2 k -v cos(~TIV)y (a -y ) 2[a+(a _y2) 2] ,y<a
6.1 Re v > -2 k 2 2 -k a y 2 (y -a) 2sin [varcsin (_) ] , y > a
y
X-~cos(ax) v+k 2 2 -k 2 2 k -v -sin(~TIv)y 2(a -y ) 2[a+(a -y ) 2] ,y<a
6.2 Re v > -1 y~(y2-a2)-~cos[varcsin(~)] , y > a
y
_3/2 sin (ax) -1 V+k 2 2 k -v X v sin(~TIv)y 2[a+(a _y ) 2] , Y < a
6.3 Re v > -1 -1 k a
v y 2sin [varcsin (-) ] , y > a y
_3h X cos (ax) -1 v+k 2 2 k -v v cos(~TIV)Y 2[a+(a _y ) 2] , Y < a
6.4 Re v > 0 -1 k a
v y2cos[varcsin(-)] , y > a y
x v-~sin (ax) TI~[r(~_v)]-12Vyv+~(a2_y2)-v-~, y < a 6.5
-1 < Re v < ~ 0 , y > a
xV-~cos(ax) v -k v+~ -v-k -2 TI 2s in(TIv) r (~+v)y (a 2_y2) 2,y< a 6.6
-~ < Re v < ~ V -h v+k -v-k 2 TI 2r(~+v)y 2(y2_a 2) 2, i' a
v+ k x 2sin (ax) -21+vTI-~a v+k -v- 312
sin(TIv)re12+v)y 2(a 2_y2) 6.7
-% < Re v < -~ y > a
_?l+v 'TT' -~;;:a r (312-4-.) \ U v+~ ('i.72_~ 2 \ -'\1- 3/2
so
f(x)
6.10
6.11
I. Hankel Transforms
= f f(x) (xy) 2JV(xy)dx o
l+v l< -1 v+l< 2 Z _v_ 3n 2 1T 2a[r(-I;;-v)] y 2(a -y ) ,y<a
o
o
-v -V-ll. 1T r(V+ll) v+1;; 2 a sln[Z(v+ll)] r(v+l) y
3 a 2 F (I;;+l;;ll+l;;v l;;+l;;ll-I;;v;-;--)
2 1 ' 2 y2
y>a
y<a
, y < a
, y > a
-v -v-ll 1T r(V+ll) v+l< 2 a cos[Z(v+ll)] r(v+l) y 2.
y < a
y > a
1.6 Trigonometric and Inverse Trigonometric Functions 51
'" 10 f(x) g(y) = f f(x) (xy) 2JV (xy)dx
° x~-V(b2+x2)-lsin(ax) -v -ab 10
~rrb e y2Iv (by) y < a 6.13
Re v > -~
6.14 1- < Re v < ~
Re v > - 5/2
xV+2n-la(b 2+x2)-1 (-1)nbv+2n-~sinh(ab)Y~Kv(bY) y > a
sin (ax)
n = 0,1,2,'"
x2n+~-v(b2+x2)-1 (_1)n~rrb2n-ve-abY~I (by) y < a v 6.17 sin (ax)
Re v > 2n- 3/2 ,n=-l,O ,1,
x2n-la-v(b2+x2)-1 (_1)nlarrb2n-v-1e-abY~I (by) y < a v cos (ax)
6.18 Re v > 2n - 5'2
n = 0,1,2, ...
00 1< f(x) g(y) = f f (x) (xy) 2J (xy) dx
o v
xV+2n+\(b 2+x2)-1 . (_I)nbv+2ncOSh(ab)y\K (by) y ~ a v . cos (ax)
6.19 -1 < Re (v+n) < 3/2_n
n = 0,1,2,'"
x- \in(ax2 ) \ 2 v+l L _\(2!Y.) sin(.l..:.. - -4- 11) J\v(Sa)
6.20 a Sa
Re v > -3
x -\cos (ax 2 ) \ 2 v+l ) L \(2!Y.) cos(.l..:..- -4-11 J\v (Sa)
6.21 a Sa
Re v > -1
x\sin(ax2 ) 1 \ 3/2 2 11 2 -11 (t.) [cos (.l..:.. - v -) J (.l..:..) -
6.22 S a Sa 4 \v-\ Sa
Re v > -4 2 11 L - sin (.l..:.. - +
Sa v4 )J\v+\(Sa) 1
x\cos (ax 2 ) 1 \ 32 2
11 L -11 (~) [cos(.l..:..- 4) J\v+\ (Sa) + 6.23
S a Sa
Re v > -2 2 11 ~ + Sin(~ - v4)J\v-\(sa) 1
x v+\sin (ax 2) -v-l v+1< L \1Iv) 6.24
(2a) y 'cos (4a -
00 k f(x) g(Y)=l f (x) (xy) 2J v (xy) dx
xV+>'cos (ax 2 ) (2a)-V-1yV+>'Sin(~ - >,lTv) 6.25
-1 < Re v<,.
xV+Y,sin(ax2 ), xb
Re v > -2 - UV+ 2 (2ab 2 ,bY)COs(ab 2 )]
XV+>'cos(ax 2 ) , xb
Re v > -1 + UV+1 (2a 2 b,by)cos(ab 2 )]
xV+Y,sin[b(a 2 -x 2 )] -v-1 V+Y, 2 (2b) y UV+2 (2a b,ay)
x<a 6.28 0 x>a
Re v > -1
xV+>'cos[b(a 2 -x 2 )] -v-1 V+k 2 (2b) Y 2UV+1 (2a b,ay)
x<a 6.29
0 x>a
Re v > -1
_(2b)-v-1 v+~v (2 2b ) Y I-v a lay
k: k; .k ~1Iy2Jv(Cb2) [Jv(db2)sin(~1Iv) +
+ y (db~)cos(~1Iv)1 + v
c k k d = (a+y) 2± (a-y) 2 y < a
h 1:: k: ~ 1Iy 2 J v (cb 2) [J v (db 2) CO S (1:; 11 V ) -
- y (db~)sin(~1Iv)1+y~I (cb~)K (db~)' v v v
• sin (1:;1Iv)
C k k: d = (a+y) 2±(a-y) 2 I y < a
1.6 Trigonometric and Inverse Trigonometric Functions
f (x)
6.36 Re v > -1
6.37 Re v > -1
= f f(x) (xy) 2JV (xy)dx o
l< l< l< ~1Ty 'Jv (cb') [J v (db 2) COS (~1TV) -
l< . - Yv(db2)s~n(~1Tv)1 -
y < a
y < a
- Yv(db~)sin(~1Tv)l
0
-31z -1 k k k x cos(ax-bx ) 2 cos (lo1TV) Y 2yv (cb 2) Kv (db 2)
6.38 Re v > -1 c k k
= (a+y) 2± (a-y) 2 I Y < a d
_ 3/2 -1 -1TyloJ (cblo) [J (dblo) sin (lo1Tv) x cos(ax-bx ) + v v
6.39 Re v > -1 + Y (dblo)cos(lo1Tv)] v
c k k
d = (a+y) 2± (a-y) 2 I Y < a
x-1o(b 2+x 2)-1o k k . -lo1TY 2Jlov {lob [a- (a 2_yZ) 2] } . 6.40 cos [a(b z+x2)1o] Y {lob[a+(a z_y 2)1o]} -lov I
y < a
x-lo(b 2+x 2)-1o k k 101Ty2Jlov{lob[a-(aZ-yZ) 2]}
6.41 sin[a(b 2+x z )1o] J lo {lob[a+(a 2_y 2)1o] y < a - v I
Re v > -1
00 k
f (x) g(y) = J f (x) (xy) 2J \) (xy) dx 0
x\)+~sin[a(b2+x2)~] ( ) ~ab \)+ 3/2 \)+~ (2 2 ) -~\)- 3/. ~~ y a -y .
-1 < Re \) < -~ {sin (~\)) J \)+3/2 [b (a2_y2)~] +
6.42 + cos(~\))Y\)+~2[b(a2-y2)~]}, y < a
- -k \)+ 3/2 \)+~ -k\)- 3/. (~~) 2ab y (y2_a2 ) 2 .
. K\)+3k[b(y2-a2)~] , y > a
x\)+~cos[a(b2+x2)~] k \)+ 312 +k -k\)- 3/. (~~) 2ab y\) 2 (a2_y2) 2
-1 < Re \) < -~ k • {cos (~\))J\)+3f2 [b (a2_yZ) 2] -
6.43
0 , y > a
X\)+~(b2+X2)-~ k k+\) k+\) -k-k (~~) 2b 2 Y 2 (a2_y2) 4 2\)
sin[a(b2+x2)~] k . .J [b(a~_y2)2] y < a -v-la -1 < Re \) < ~ 0 > y a
0
XV+~(b2+X2)-~ (' )~b~+v ~+v( 2 2)-~-~V ~rr y a -y
k sin[a(b 2+x 2)l:2]
. J [b(a2_y2)2] y < a 6.44 -v-~
-1 < Re v < ~ 0 y > a
xv+l:2(b 2+X2)-l:2 . k k+ ~ + -k-1 V _ (l:2rr) 2b 2 Y V (a2-y2) 4 ~
cos [a(b 2+x2)l:2] k . Y [b(a2-y2) 2] y < a
6.45 -v-~
k K [b(y2-a2)2] y > a v+~
xv+l:2(k 2+x 2)-1 v k k: . k y 2cosh[a(k2-b 2) 2]Kv (kY) y > a
6.46 cos [a(b 2+x2)l:2]
-1 < Re v < 3/2
xv+l:2(k2+x2)-1 V k _k: k: k y2(k 2_b 2 ) 2sinh[a(k2-b 2) 2] .
'(b2+X2)-\ . Kv(ky) . 6.47 y > a
sin [a (b 2+X2) \]
-1 < Re v < 5/2
6.48
6.49
6.50
_x-l:;(x 2 _a 2 )-l:;
x>a
= J f(x) (xy) 2J\I(xy)dx o
k \I \I+k 2 2 -k\l-h (l:;a1T)2a y 2(b +y) 2 4
S9
= f f(x) (xy)~J (xy)dx o v
-~ ~ v+~ 2 2 -~v-~ v+~ 2 TI a (b +y ) y •
y [a (b2+y2)~] v+~
x < a
x > a
Re v > -1
o x < a
l< - v l<-v 2 2 l<\)-l< 2 2 l< (~TIa) 'a y 2 (y -b ) 2 4JV_~ [a(y -b ) 2]
6.52 Y > b
Re v > -~
o x < a -~ v+ 31z . ( ) v+~ (b 2 2) -~v- 3/. -(~TI) a s~n TIV y -y
y a
• yv+ 312 [a (y -b ) 2] y > b
f(x)
= J f(x) (xy) 2JV (xy)dx o
-~ v+ 3/2 V+k 2 2 -kv- 3/4 - (~'1T) a sin ('1Tv)y 2(b _y ) 2 •
0
k 312_v k-v ~v- 3/4 (~1T) 2a by 2 (y2_b 2 )
k . J V_3J2 [a(y2_b2) 2)
k • K, {~a[b+(b2_y2)2)}
'liV
• Iv(ky)
I
0
0 x < a ~~y~k-vexp[-b(a2+k2)~lIv(kY) > b y
x~-v(k2+x2)-1
Re v > _ 3/2
X~-V(b2+X2)-~ . yV+~[a+(a2_y2)~1-v(a2_y2)-~
• [ (b 2+X2) ~-bl v k . cos [b (a 2_y2) '+~~vl y < a
6.58 sin[a(b2+x2)~1 y-~(y2-a2)-~exp[-b(y2-a2)~1 ·
Re v > -1 sin[v arcsin (~)l y > a y
X~-V(b2+X2)-~ . v+k 2 ~ -v 2 2-k -y 2 [a+ (a _y2) 1 (a _y ) 2
[(b2+x2)~-blV k . . sin[b(a2-y2) 2+~~vl y < a
6.59 cos[a(b2+x2)~1 -k -k ~ · y 2(y2_a 2 ) 2exp[-b(y2-a 2) 1 .
Re v > -1 cos[v arcsin (~) 1 . , y > a y
'" k f (x) g(y) = J f(x) (xy) 'Jv(xy)dX
0
-1 -axJ-. -k -k X e DV-J-. (ay 2) D_V-J-. (ay ')
6.60 . cos[axJ-.-J-.~(v-J-.)l Re v > -J-.
x-J-.(a 2 _x 2 )-J-. J-.~yJ-.JJ-.V+J-.~(J-.aY)JJ-.V_J-.~(J-.ay)
6.61 . cos[~ arccos(~)l a Re (v+~) > -1
64
2 -2 • exp(-'/sY a )
1.7 Orthogonal Polynomials
7.5 Re (2n-v) > _ 3/2
V+>< 2 x 2exp (-a 2x ) . 7.6 LV(a 2x 2)
n
x v+~exp (-~a 2X 2) . 7.7 LV(a 2x 2)
n
7.8
(n!)-12-na-n-V-1yn-~exp(_1~y2a-2)
• YV+~Lv(y2a-2) n
Re v > 0
l< 2 [L 2V (ax 2 ) )
n
n
Re v > -1
o x < a
2n-y, < Re v < 2n+y,
-v-1 v+Y, 2 -1 (2a) y exp(-~y a ) •
l< 1 2 [L~v(~2a- »)
-1 -v-1 v+Y, (TIn!) r (n+l+y,v) (2b) y •
(_l)mr (n-m+y,)r (m+y,) • r (m+l+J;v) (n-m) !
(_1)n 22n-v+1 r(2v-2n)'
2n+ 3/2 -~ 2 2 v- 2n- 312 x \x -a ) •
v-2n-1 -1 • C2n+1 (ax )
x > a
x < a
Ox> a
. cos[b(a2-x2)la] . ·cv+la [(1_x2a-2 )la]
2n
o
(_1)n(laTI)la(ay)v+la(b2+y2)-lav-~ •
(_1)n(laTI)la(ay)v+la(b2+y2)-lav-~ •
67
0
Re v > - ~
XV+~Ei(-ax2) -2 V+ly -v- 3/2 y(v+l,l:oy2 a
-1
+;. 2 -r(l+v) (2a)-V-lyv+~exp(l:oy2a-l) XV 2eax Ei (-ax2 ) .
B.3 -1 < Re v < 3/2 2 -1 . f(-v,J..oy a )
v- k 2 2 x 2exp(a x ) 11- ~a-vr (\+v) y -~ 1 2 - 2 exp ( lay a ) . B.4 Erfc(ax) . W (l:oy2a- 2 )
-v,v -~ < Re v < 3/2
x~+vErfc (ax) a-v r(31z+v) [r(v+2»)-ly _ 3/2 .
B.5 Re v > -1 lA 2 - 2 2 -2
'exp(- ay a )M~+~,~v+~(J..oy a )
x -v-lfiy(v+~,ax2) -v ~ v-\; -1 2 11 Y rfc (J,;ya )
B.6 Re v > -~
2-v-1 v+\;. ( 1 2 -1) x - Y ~ -~y a B.7
Re v > -1
0
xV+~xp (ax 2) r (-v ,ax2) -[r(1+v)]-1(2a)V+lyV+~xp(ax2)Ei(-ax2) 8.8
-1 < Re < 312
X~-V exp (-a 2X 2) (2a2)v-ly~-vexp(_\y2a-2) . 8.9 L_v (a2x 2) · L_ v (\y2 a - 2 )
Re v < ~
k+v x 2 exp(-a2x 2 ) [r(~+v)]-12-2va-2V-lyV-~.
8.10 L (a 2x 2) 1 2 -2 -~ -1 . · exp(-/eya )D2v (2 a y) v-~
Re v > - ~
k+v • [r (3/z+v ) ]-12-2v-3/2yv+~exp(_1/ey2a-2) x 2 exp(-a 2x 2) . 8.11 L (a 2x 2)
v+~ -~ -1
Re v > -1
~-v 2 2 x exp(-a x ) . k -1 -v -1 v-k 1/, 2 -2 1T 2 [f(V)] 2 a y 2exp(- By a ) .
8.12 . LV_l(a 2x2) · I 1 (1/ey 2a- 2 ) \)-'2
Re v > 0
'" k f (x) g(y) = f f (x) (xy) 2JV (xy)dx
0
lztv exp (-a 2x 2) -J" -1 v -1 -v-J" 1~ 2-2 . x 1[ [r(-v)] 2 a y exp(- Sy a )
8.13 · L- v - 1 (a 2x 2) K efi y 2a -2) v+J"
-1 < Re v < - 1/6
xJ,,-vexp (_a 2x 2) . [f(V+k+1)]-12-ka v-k-ly k-J"
8.14 · L (a 2x 2) · exp(_I/sy 2a - 2 ) . k
Re(v+k) > 0 · 2 -2 M~_l:>v+J",J.-"k+J"v (>.,y a )
k+v x' exp(-a2x 2) [f(1+k)]-12-ka -k-V-1y k-J"
· Lk (a 2 x 2 ) · exp (_I/sy2 a -2)
8.15 W (1/6y 2 a -2) Re(2k-v) > - 3iz
J"k+J"+J"v,J"k-J,,v
Re v > -1
1.9 Legendre Functions
1.9 Legendre Functions
• p -V-~ (2a 2X -2_1 ) \1
x < a
Ox> a
x < a
J. 1-\1 \1 ('orr) 2a y JJ.- (~y)J (~y)
2 \1 v
• F (V+\1+3/2 ;2V+2;-iay) 1 1
= (271) -~[r (3f2+V) ]-2r (3/2 +\1+V) r (~+v-\1) •
• 2 v+% V-\ -V- 312 •
~ -v-1 V+~ 71 2 ay JJ. (ay)J '+ (ay)
2-\1 -,. \1
o x < a
9.5 x > a
o x < a
9.6 x > a
2 2
Re(v±lI) > -2
• P>, 11->' (I 2 2 I ) a -x
Re 11>-1, Re(II-2v)<1
00
-v-2 3/2 -1 2 TI a[f(l+v-II)f(l+v+II)] •
V+k 2 2 • y 2{[J II (>,ay)] +[YII(>,ay )] }
• y-II->'K (ay) v
f(x) (XY)~Jv(XY)dX f(x) g(y) = f 0
x~(a2+x2)~V . 21+v -1 ~ a COS(~~~)y-V-~K~(ay)
9.12 • P~~_~(1+2x2a-2)
-1 < Re ~ < 1
x~(a2+x2)-~V . 21-va[r(~+v+~~)r(~+v-~~)]-1 . 9.13 -v 2 -2
• p~~_~(1+2x a ) · v-~ y K~ (ay)
Re(2v±~) > -1
xV+~(a2+x2)~V . (2a)V+1[~r(_v)]-1 . -v-~ 2
9.14 2a 2+x2 • y [Kv+~(~ay)] • p [ ] v 2a(a2+x2)li
-1 < Re v < 0
v+~ x 2+2a2 (2a)V+1[r(_v)]-1y-V-~ . x pv [] 9.15
2a(x2+a2)~ · I_v_~(~aY)Kv+~(~ay) O<Rev< 312
1.9 Legendre Functions 75
f(x) (xy)~J (xy)dx f(x) g(y) = 6 \I
x\l+~[ (a+2b2x2)~lC%\I-~. n-~2-2\1-~-2\1-2[r(1+\I+~)r(1+\I_~)]-1 . • p-\l-~(a+2b2x2) • y\l+~K {2- 3/2b -ly [(a+1) ~+ (a-1)~]} .
~-~ ~ 9.16
Re \I>-l,Re(\I±~»-l .K {2- 3Izb -ly [(a+l) ~- (a-1)~]) ~
a ~ 1
x~(a2+x2)-~ . 2~-\la~[r(1+~\I+~~)r(~+~\I-~~)]-1 . • p-\l[(1+x2a-2)~]
~ • Kl.+ (ay) . ~
Re(\I+~) > -~
X~~~~[(1+x2a-2)~] 23~-\la~[r(~+~\I+~~)r(~+~\I_~~)]-1 . 9.18 -1 Re (\I±~) > - ~ • Y K~ (ay)
x~{p-~\I[(1+x2a-2)~]} 2 2an-1[r(~+~\I+~)r(~+~\I-~)]-1 .
~-~ 9.19
xJo(a 2+x2)-Jo a~-l[r(l+Jov+jl)r(l+JoV-jl)]-l •
• p-JoV-Jo[(1+x 2a- 2)Jo] •• yJo[Kjl(Joay)]2 jl-Jo
9.20
Re(Jo±jl) > -1
-"V 2 -2 " P 2 [(l+x a ) 2] • -jl 9.21
Re v>-l,Re(Jov±jl»-l
9.22
1.9 LegendTe Functions 77
o
Re v > -1, Re p <~
Re(v+p) > -1
-1<Rev<O
9.26 iu( ~-v) v-~( 1+2 2 -2) "e q_~ a x
o < Re v < 3/ 2
-k -v-l k+v u'2 ay' Ip (~ay) Kp (~ay)
9.27
78
9.29
9.30
9.31
9.32
9.33
f(x)
• e-i'll'(V+~) •
Re(V±l1) > -1
i'll''' _II 2 -2 ~ e '"q '" [(l+a x )'2] v-~
Re (v±l1) > -1
• ql1 [(l+a2x-2)~] v-~
~ -2V- 3/2 -2v-2 ~+v 'II' 2 by'
• I {2- 312b -ly [(a+1) ~- (a-1)~]} • 11
• K {2-3kb-1y[(a+1)~+(a_1)~]} 11
'II'~2-V [r (3/4+~V+~11) r (3/4+~V_~11)] -1.
• r(~+V-l1)Kl1(ay)
-1 • r(~+v-l1)Y Kl1 (ay )
1.9 Legendre Functions 79
co k f (x) g(y) = f f(x) (xy) 2JV (xy)dx
0
x-~(a2+x2)-~ei2n~ r(~v-~) k . 2r (l+~v+~) Y"K~_~(~aY)K~+~(~ay)
k-~ -2 k · q" [(1+a 2x) "]. ~v-~
9.34 q-~-~[(1+a2x-2)~] · ~v-~
x-~ei2n~ . -~ r(~+~v-~) [K~ (~ay) ]2 y r (~+~v+~)
9.35 · {q~ [(1+a2x-2)~]} ~v-~
Re(~±~) > - ~
-k -~ 2 -2 ~ y-~I~(~aY)K~(~ay) x"p [ (l+a x ) ]. ~v-~
9.36 -in~)J 2-2 k · e q [(l+ax )"] ~v-~
Re v>-l, Re (v+2)J) >-1
x~p-~V[(1+x2a-2)~] . aY-~I (~ay)K (~ay) )J-~ )J ~ .1[
-~2v kv -2 ~ 9.37 .e q2 [(1+x 2a ) ]
)J-~
80
10.1
10.2
-1 < Re(v+ll) < 0
x>,,+Il- VJ (ax) 10.3 Il
Re (WIl) > -1
o
• (a 2_y 2)-Il- v-l y < a
-1 v+Il+1 . Il v+k -~ 2 s~n(~Il)r(ll+v+l)a Y •
y > a
2V- Il+1 -)1 V+>" 2 2 Il-v-l r (Il-v) a y (a -y ) , y < a
o , y > a
o , y < a
21l- v+1 Il ~-V 2 2 V-Il-1 r(v-Il) a y (y -a ) ,y > a
v (>"+v->"A) 2-1. Y r(>"A+>,,) •
• (la 2_y 21)>"A->"p-V (a 2+y2 ) ~A->" 1 a 2_y 21
=2-A(rra)->"[r(>"+~A)1-1~a2_y21>"A •
i2!.A • e 2
10.5
10.6
10.7
10.8
f(x)
00
y < a
y > a
y > a
Eason, G. Noble, B. and Sneddon,
I. N., 1955:
Re v > -1
= f f(x) (xy) 2JV (xy)dx o
L 2 2 _L).I_k -kV • y'2 (a _y ) '2 4 p ' ( z ) =
).I-~
-k 2 2 -k).l -i~).I ).I 2 -2 • Y '(a -y ) • e q (2a y -1)
~v-~
y < a, 2 '-2 -2 -k z = (1-~ a ) (1-y 2a ) •
2).1r (~+~v+).I) [r (~+~V-).I) l-ly-~ •
• (y2_a2)-~).Ip-).I (1_2a 2y-2) ~v-~
• (a2_y2)~).Iq).l (2a2y-2_1) ~v-~
1.10 Besse;!. Functions of Argument x
f (x)
Re \! > n-1
l+Re ~ > Re \! > -1
Re ~-2n+1 > Re \! >
10.17
y~a
y ~ a
y ~ a
y < a
y < a
f (x)
(-I)ny~I (by)K 2 (ab) v v- n y < a
10.18 n = 0, 1, 2,··· (_I)ny~I 2 (ab)K (by) v- n v y > a
Re v > n-I
x~(b2+x2)-ly (ax) (-I)ny~K 2 l(ab)I (by), v-2n-l v- n- v y ~ a
10.19 Re v > n - ~
10.21 +sin[~TI(p-~+v)]Y (ax)) ~
Re p < 1
Re(v±~+p) > -2
y ~ a
f(x)
Re p<3/2 , Re(p+Il»-2
Re(p+Il+2v) > -2
'{COS[I(P+O-Il)]JIl(ax)+
i=l
Re(p+o»IRe 111-2
o _2~-ly-~(y2_4a2)-~
y < 2a
y > 2a
y < 2a
-~ 3v+1 2v L ~ 2 a -v-~ 2 2 -V-'ll r(~-v) y (y -4a ) ,y>2a
CPY~IIl(bC)IV(CY)KV(aC) Y < a - b
k Y < a - L c i
i=l
10.26
-A A-v-l 2 a
• r(~+~V+~]1-~A)r(~+~V-~]1-~A)y~+v •
-1 cof -A-~ y. =~ sin[~~(v-]1-A)l x IV(XY) (xy) 'K\l(ax)
o dx
y < a
co
f -A-Y. ~ cot(~]1) x 2J (ax) (xy) J (xy)dx o ]1 v
COf -A-Y. Y. -csc(~]1) X 2J (ax) (xy}'J (xy)dx o -]1 v
Y > a
Ch. II) for y > a
1.10 BesseL Functions of Argument x 87
00 l< f (x) g (y) = f f(x) (xy) 2J (xy)dx
0 v
xl:i+v+j.ly j.l(ax) -1 v+j.l+l v+l< j.l TI 2 cos(TIv)r(j.l+v+l)y 2a .
-l<Re(v+j.l) < 0 10.28
(a 2_y2) -j.l-v-l , y < a
Re v > -1 -1 v+j.l+l j.l v+l< - TI 2 COS (TIv)r(j.l+v+l)a y a . . (y2_a 2)-j.l-v-l
, y > a
xl:i+v-j.ly j.l(ax) -1 v-j.l+l -j.l v+l:i -TI 2 r(v-j.l+l)a y . Re v > -1, (y2_a 2) j.l-v-l , y > a
10.29 -1 < Re(v-j.l) < 0 -1 V-j.l+l -j.l v+l< TI 2 -cos[TI(j.l-v)]r(v-j.l+l)a y a
. (2 2)j.l-v-l Y < a -y , a
xl:i[Jl< (ax)] 2 -1 -l< 2 _l:i = 2TI y 2 (4a _y2) y < 2a aV
10.30 0 2a Re -1 y > v >
l:i -1 -2j.l _l< -l< x Jl< + (ax) TI (2a) y 2(4a 2_y2) 2 .
2V j.l
aV-j.l -1 -l:i -l< -1
10.31 Re v -1 = 2TI Y (4a 2_y2) 2COS [2j.larccos (l:;ya ) ] >
y < 2a
Re(v+J.I) > -I,
10.34 Re p>-l, Re(p-J.l-V)<~
b > a > 0
x~-vJ (ax)J (bx) v v
Rev>-~
10.36
g (y) = I f(x) (xy)~J (xy)dx o v
n+1 2n-J.l-2 ~ (-1) e y I (be)l (ye)K (ae) J.I v v
y < a-b
sides a, b, y
00 1<
f (x) g(y) = f f(x) (xy) 2Jv (xy)dx 0
~-v V(ax)JV(bX) 0 la-bl and y > a+b x J , Y <
Re v > - ~ 1I-~(ab)-v21-3v ~-v r(!li+v) y
10.37
x -~ [J (ax) 1 r(~+~v+].I) . ].I r(!li+!liv-].I)
10.38 Re(v+2 ) > -1 -~ -].I 4a 2) ~12 } , 2a . y {p -~+~v [ (1- Y >
y2
X-~J ].I (ax)J_].I (ax) y-~p~ ~[(1_4a2)~lP~].I ~[(1-4~)~1 10.39 2V- y2 v- y2 Re v > -1
y > 2a
~ -1 -~ -1< ~ x J~(v+n) (ax)J~(v_n) (ax) 211 y 2(4a 2_y2) 2T ( ), o < y < 2a 10.40 n 2a
Re v > -1, n=0,1,2, ... 0 , y > 2a
~-v 2
y v-~ (4a 2_y 2) v-~ , <y< 2a Re v > - ~
r (!i+v)
0 < y > 2a
10.42
jJ -jJ
~+v x J (ax)J (ax) v -v
10.46 -1 < Re v < ~
o < y < 2a
o < y < 2a
o y > 2a
-~ I-v -1 2 l<v-~ -TI 2 a sin (TIjJ) (y2-4a ) 2 •
• ~-v (y2 1) PjJ-~ ---
• y-V-~(4a2_y2)-V-~,
o
f(x)
Re ].I > -1, Re \I > - ~
10.47
10.48
91
o o < y a + b
• (Z2_1)-~\I-~e-i~(\I+~)q\l+~(z ) 1 ].1--. 1
y < la-bl
(2~)-~(ab)-\l-ly\l+~(I-z2)-~\I-~ • 1
• [p\l+~(z )COS(~\I)- l Q\I+~(z )sin(~\I)] ].1--. 1 ~ ].1--. 1
I a-b I < y < a+b
-(~~3)-~sin(~].I) (ab)-\l-ly\l+~ •
• (z2_1)-~\I-~e-i~(\I+~) \I+~(z ) 2 q].l-~ 2
Zl as in 10.47 2
Y > a+b
0 v
x-~[J2(ax)+y2(ax)] -1 J" 2 (7TY) esc (7T].I)Y 2 . ].I ].I
Re(v±2].1) > -1 4a 2)~] 0-].1 [(1_4a2)] . {P].lJ,,+l [(1- -
-. 'l;V y2 -~+~V 2 Y
10.49
_ P -].I [ (1- 4a 2 ) ~] 0].1 [(1_4a2)~]} -~+~V y2 -~+~V 2
Y
Y > 2a
xV+~[J2(ax)+y2(ax)] 2V+1 7T-o/2a-1r(1+V+].I)r(1+V_].I) . ].I ].I
10.50 Rev < ~, Rev > -2 . (y2_4a2)-~-~Vp-V-~(~ -1) y > 2a ].I-~ 2a2
Re (v±2].1) > -2
1.10 Besse,l Functions of Argwnent x
f(x)
-1 < Re v < ~
Re(v+jl) > -1
-1 < Re v < l!j
10.52 Re(V+Jl) > -1
(~7f3) -~cos (7fv) (ab) -v-1y v+~ •
• ( 2_1)-l.i-~v -i7f(v+~) V+~(Z ) Zl e qJl-~ 1
Y < a-b
a-b < y < a+b
y > a+b
Y < a-b
as above
x -lzJ (ax sin C4 cos 13) • Jl
Jp(ax)
• JV_Jl(ax}
0= I Jl i i=l
-1< Re v< Re o+J.-ak-lz
o v
Math. Soc. (2), 40, 37-48.
(sin C4 cos I3)Jl (sin 13· cos C4)v+lz •
·2Fj (lz+lzo-lzp,lz+lzo+lzp;1+v;sin 2 13)
o = Jl+v, Y = a cos C4 sin 13
-1 _3(2 Jl V+'-' 21T a sin (1TJl) (sin a) (sin 13) 2
• [cos (a+l3) - cos (a-l3) ]-1
y = a cos a sin 13
o
f (x)
k 10.58 6 = L lli
i=l
95
o
f (x) g(y) k
xloJ (ax 2 ) lov
11.1 Re v > -1
11.2 lov
2V 2 11. 3
11. 4 aV- 2
Re v > - 5/2
(l,;ax 2 )
co
f(x) (xY)~J (xy)dx f(x) g(y) = f 0 v
x~ [JItV(JolaX 2) ] 2 -1 ~ x.: :C. -a y J ( ) Yltv (4a) ltv 4a
11. 7 Re v > -1
X~Jltv(ltax2)Yltv(ltax2) -a -1y~ [Jltv (li) ]2
11.S Re v > -1
• 'IT
X~J (ax2) -1 - 3/2 14" . 2 'IT Y [e W It (z)W It (z ) + Itv-lJ lJ, v 1 -lJ, v 1
11.9 • J It + (ax2) .7[ -17'\1
V lJ - + e 4 W lJ,ItV(Z2)W-lJ ,ltv(Z2)] Re v > - ~
2 ±i.!L Zl = ~ e 2
Sa 2
11.10 Rev>- ~
Re v > ~
_ 3/2 -1 a-~J2V_1[2(ay)~] x J V_1 (ax ) 11.12
Rev>-~
98
11.13
11.14
11.15
-Re(v+ 312 )<Re p <
Re(p+v) > - 10
Re (p-v) < 3/2
~ +B·oF3(1+v,1+~V-~\.1r~p,1+~\.1+~v+~p; 16 )
2P-1-2\.1a\.1y\.1-P+~r('v+~e-~\.1) A = r(1+\.1)r(1+~v+~\.1-~p)
2-p-1-2vav+pyV+~r('H_;V_~P) B = r(v+1)r(1+~v+~\.1+~p)
V-~ -~i CSC (2TTV) (~) •
2
2
2 -y-\.1BoF3(1+\.1,1+~\.1+~v-~p,1+~\.1-~V-~p;ti)]
A-1=2v+Pr(1+v)r(1+~p-~\.1+~v) •
• r(l+~p+~\.1+~v)
• r (l+~\.1-~v-~p)
1 . 11 Bessel, Functions of Other Arguments 99
00 k
f (x) g(y) = J f (x) (xy) 2J (xy) dx 0 v
• ~ -k -a 2b
x (b 2+X 2) 2exp (-----)· -~ -by J 2v (2ay~) b 2+X2 y e
2 11.16 . J (~)
Re v > - ~
~ J 2v- 1 (ax) ~ay _3/2 2 -1 J v _ 1 (>;;a y )
11.17 Re v > - ~
Y 2J V (>;;a y )
11.18 Re v > - ~
-k -bx k k 2 2-k a 2b x 2e J 2V (2ax 2) y 2 (b +y ) 2exp ( - --) . 11.19
b 2+y2
100
Re v > -1, Re(v+ll)<O
11.21
11.22
11. 23
g(y) co k
a -Pb V-ll+1 (a2_y2) ~Jr~v-\v+~ •
J [b(a2-y2)~1 p-v-1
o y > a
llbV+Jl+1 v+~ (2 2)-~Jl-~\!-~ a y a -y "
-COS(~V)Jll+V+1[b(a2-y2)~1} y ~ a
_2~-1sin(~1l)allbv+ll+1y\!+~ •
y > a
y > a
y > a
11.24
v+2n- 3f2 (2 2-1 x a +x) -
n = 0, 1, 2,---
v+~ 2 2 -~v- ... x (b +x ) -
- C~~11[b(b2+X2)-~]-
v+~ 2 2 -~v- ... x (b +x )
n = 0, 1, 2,---
y ~ c
y < c
( l) n(, )-~ -~-v v+~( 2 2)-~ - •• a y a -y -
~ v+~ y2 ~ - sin[b(a 2-y2) ]C2 +1[(1---) ],
n a 2
o , y > a
V+k n -~. x 2 n z. 1 J~ (a.z.)
i=1 1 i 1 1
n ~n+ L ~i-~>Re V>-1
i=1
g(y) k
n 2 v-lb-n~r(v)y~-V n (b)
i=1 J~ a i
n y > L
i=1
v_ 3/2 nn -~i V-I ~-v x z. J~ (a.z.) 2 r(v)y •
i=1 1 i 1 1
n ~n + L
i=1 y >
11.31
11.32
o x < b
o x < b
all+V+lbllyV+~(b2+y2)-~Il-~V-~ •
!. • J [a (b 2+y2) 2] ll+v+l
o y < a
• J [b(y2_a2)~] V-ll-l
• K [b(a2_y2)~] ll+v+l
llbv+ll+1 ~+v( 2 2)-~Il-~V-~ a y y -a •
• {Sin(Wll)Yll+V+l[b(y2-a2)~] -
- COS(Wll)Jll+V+l[b(y2-a2)~]}
o x < a
n=0,1,2,'"
o v
-A ~-v d m d n b y (bdb) (ydy)
• (b2+y2)-~(A+v+m+n+1) •
( 2 2 ~ • J A+v+m+ni-l [b +y ) 1
y 1
-)lbV-)l+l v+l:2( 2 2)l:2)l-y'V-y. a y a -y •
• Y [b(a 2_y2)l:2] y < a )l-v-l
Re )l > Re v > -1 2 -)lbv-)l+l v+l:2( 2 2)y')l-Y.V-l:2 - TI a y y -a •
y > a
)lbV+)l+l v+l:2( 2 2)-y')l-~V-~ -a y a -y •
h • {cos (1fV)Y)l+v+l [b (a 2_y2) 2] +
Re v>-l,Re(v+)l) < 0 . h + s~n(1fv)J)l+v+l[b(a2-y2)2]), y < a
-1 )l v+)l+l v+h - 21f cos (1f)l) a b y 2
y > a
00
f (x) (xy)~J (xy)dx f (x) g (y) = f 0 v
xV+~J {b[(a2+x2)~+x]}' 2-2~aPb-P[r(l+p)r(p_v)]-1 . P
11. 41 .J {b[(a2+x2)~_x]} .yv+~(4b2_y2)p-v-l . p
2 -1 Re p > Re v > -1 'lF2[~+p;2p+l,p-v;-(ab-~ay b )]
y < 2b
0 Y > 2b
k -1 ~ y2 x 2y (ax 2) -~a y H~v (4a:) 11.42 ~v
Re v > -1
11. 43 -~a-lY~[J~v(1:a)]2
Re v > -1
X-~y -1 -1 -~ k k v(ax ) -21T Y [K2v (2a"y 2) - 11. 44
-~ < Re v < ~ -~1TY2v (2a~y~)]
_ 512 x y -1
2y'(a1T) [K2V (2a y2)+~1TY2v(2a"y2)]
11. 45 ~<Rev< 512
11.46
Re v > -1
-~ a 2 2 sec(Trv)y [~cos(TrVlYv(4yl-
, y < 2a
-1 -~ 2 2 -~ -b(y2-a2l~ -2Tr y (y -4a l e
Y > 2a
f(x)
12.1
Re v > - \
Re v > -1
12.6 K]1(ax)
Re(v±]1) > - 3/2
'b-1 \ (a 2_y2 ) J (ay) '2 y exp ~ v 2b
= 2 v 1T -\ r ( 31'++\V+\]1) r ( 3/4+\V-\)1) • r(\+v-]1)
1.12 Modified Bessel Functions of Argument x
f (x)
Re \I > -1, Re (v+211»-1
x±]l+\I+I:;Kll(aX)
-1 < Re ]l < 1
= 2v-1n-1:; r (~+I:;\I+l:;u)r(\+I:;\I-l:;u) • r(I:;+\I-jl)
in" -" _a2 ) 1:;] • e "q" [(1+ v-I:; y2
109
110
12.11
12.12
12.13
12.14
-v 2 -2 • p~jJ_~(1+2y a ) =
= 1T -~2 v-I f( ~+v-~jJ) (a 2 +y 2 ) -~V-~ •
2-1-AaA-V-l[r(I+V)1-lr(~+~V+~jJ_~A)
• r(~+~V-~jJ-~A)Yv+~ •
00
f(x) (xY)~J (xy)dx f (x) g(y) = J 0 v
x-A-~K II (ax) -A-1 A-1-v v+~ 2 a :t:' r(~+~V+~ll-~A) . r (l+v)
12.15 Re(v±ll-A) > -1 · r (~+~V-~ll-~A) . '2Fl (~+~V+~ll-~A,~+~V-~ll-~A;l+v;
y2 - -) a 2
12.16 'K~ +~ (ax) · [ (4a 2+y2) ~+y]ll v 2ll
Re v>-l, Re(1l-v)<2
x~r~ (ax)K~ (ax) .v v
y-~(4a2+y2)-~
12.17 Re v > -1
x->Or (ax)K (ax) ->0 e-i~1-Ip-ll [(1+4a2y-2)>o] . II II
y >ov->o
12.18 Re (2ll+V) > -1 1-1 2 -2 ~ · q [(l+4a y ) 2] >oV->o
Re v > -1
x->Or -1 >0 -~-v
II (ax)K1-I (ax) (2a) y e 2 . 12.19 Re v>-l, · p ->ov [ (l+L) >0] q>Ov [(l+L) >0]
ll->O 4a2 ll->O 4a2 Re(v+2 ll »-1
'" h f (x) g(y) = f f (x) (xy) 2J\) (xy) dx
0
h x 2KO (ax) J v (bx)
h -1 V -v y2(Z l Z2) (Z2- Z1) (Z2+ Z1)
12.20 Re v > -1
= [a 2 + (bW) 2] ~ Zl 2
v+h x 2JV_ 1 (ax)Kv _1 (ax) rr-~23V-la2v-2r(~+v)yv+~2(y4+4a4)-v-~
12.21 0 < Re v < ~
XV+~I v(ax)Kv(bx) rr-~23V(ab)vr(~+v)yv+~ . 12.22 ..,
2 2 -v-~ Re v > -~ [(b 2+y2_a 2) .. + 4a y ]
]
12.23 'Iv_~(~ax)Kv_~(~ax) v -~ v-~ -v-~ = 2 rr a y
0 < Re v < 312 (a2+y2)-~eirr(~-v)qV-~(1+2a2y-2) .
-~
\l+h -~ -1 -\l-~ -(\l-~v+~)rri x 2r (ax) K (ax) (2rr) aye v \l
12.24 Re v>-I, Re \l <~ (1+ ~)-~\l-~ \l+~ 4a 2 qv-~ (iL)
2a
'" I< f (x) g (y) = J f(x) (xy) 2JV (xy)dx
0
v ~ (27T) 2a- b y 2(z2_1) 4 2
12.25 Re v>-l, Re(v+~»-l e-i7T(Ya±~) ~±~(z) qv-~
z = a 2+b 2+y2 2by
xv-2~+YaI (ax) K (ax) 2v-2~ r(v-~+l) ~ ~ .
r(~+l) 12.26 Re v>-l, Re(v-~+l»O
Re(v-2~) < Ya 2~-V-'2 F (v- +1 y 2 I ~ , ~i ~+1; 4a 2 - -)
y2
xYa+~J v(bx)K~(ax) Yaa~b-~-lr(v+~+l)y-~-~
12.27 Re v>-l, Re(v+~»-l -I<~-I< -v 2-1< . (z2-1) 2 2p [(z) (z -1) 2] ~
z = a 2+b 2+;t2 2by
v+ k x 2J v(aX)Kv(bx) 7T-~23v(ab)vr(Ya+v)yv+Ya 12.28
[(a 2+b 2+y2)2_4a 2y2]-V-Ya Re v > -~ .
xV+YaJ v_1(ax)Kv_1 (ax) 7T -Ya2 3v-1a 2 v- 2r (Ya+v)y v+ 51z(y "+4a " ) -v-Ya
12.29 0 < Re v < Ya
114
12.30 a > b
12.31
Re(p+v) > -1
(2n)-~(ab)-V-1YV+~(Z2_1)-~V-~ •
• p-j.l[z(z2-1)-~] v
• (z2_1)-~v-~p-v-~(z) = j.l-~
einj.lq-j.l[z(z2_1)-~] v
y+ib = i a cot(~o+i~cr)
1.12 Modified Bessel Functions of Argument x 115
f (x)
12.34 Re v>-l, Re(v+~»-l
b > a
-1 < Re v < 0
Bailey, W. N., 1936: Proe. London
Math. Soc., (2)40,37-48
~ -v -v-~ v-~ 1T 2 a y
• (a 2+y 2) ~v-"'ei 1T (v+~) q -v-~ (1+2a 2y -'2) -~
21Ti" -" 2 -2 L -" -2 >< 'e '"'q '"'[(l+ay )'2]q'"' [(1+a2y )2] ~v ~v-1
-1 >< ->< = 1Ta y2r(1+~v+~)r(1+~v-~) (a 2+y2) 2 •
• p-~v-~[(1+y2a-2)~]p-~V+~[(1+y2a-2)~] ~-~ ~-~
f(x) (XY)~J (xy)dx f(x) g(y) = f 0 v
X-~[K (~ax)]2 r(~+~v+ll) y-~e2will{q-1l [(1+a2y-2)~]}2 Il
Re(~v±ll) > - ~ r pi+Jiv-ll) ~v-~
12.38 -1 ~ = ~wa Y r(~+~v+ll)r(~+~v-ll) .
· -l§v {PIl-l§ [(l+y2a -2) l§]}2
Xl§K 1l_~(l§ax)KIl+l§(~ax) -1 -l§ wa r(l+l§v+ll)r (l+l§v-Il) (a 2+y2) •
Re v>-l,Re(~v±Il»-l • y~-~v[(1+y2a-2)~]p-~v[(1+y2a-2)l§] -Il Il
12.39
y-l§(a 2+y2)-l§ •
.ei2Wllq~-1l [(1+a2y-2)~]q-l§-Il[(1+a2y-2)~] l§v-~ ~v-~
XAK ll(ax)Kp (bx) Bailey, W. N. , 1936: Proe. London 12.40
Math. Soc. (2) 41, 215-220
X-v-l§[Kv+~(~ax)]2 -v-1 v+~ 2 2 ~v (2a) wr(-v)y (a +y ) . 12.41
2a2+~2 -1 < Re v < 0 P [] · 2a(a 2+y2)l§ v
1.13 Modified Bessel Functions of Other Arguments 117
1.13 Modified Bessel Functions of Other Arguments
00
0
l<-v -l< -1 v-l< L)D -2v (~) x' exp(-\a Zx 2) . (~11) 2a y 2exp (- 4a z
13.1 · I (\a zxz) v
Re v > - ~
v-slz -l< v+l< z.: (:i.) x exp (_\a Zx 2 ) (~11) "y 2exp (-\ ) D_ 2v- 3 a Z a
13.2 · I (l<a Zx 2) v+1 4
Re v > -1
l< -l< x..:. x 2exp(-\ax 2)Il< (\axz) (~1Iay) 2exp (- 2a) 2V 13.3
Re v > -1
lis (v+l<) X 2 exp (-\ax 2 ) -1 _lis (v+2) lis (v+~) (ky2)
11 a y exp -4a- . 13.4 · I (+l<) (\ax 2 ) 2
l/S v • . K (\:c.) l/S (v+~) a
-1 < Re v < 5/2
_1/3 (v-I.) x 2 exp(-\ax 2). l/S (v - 2 ) lis (~- v ) (k:t.:.)
a y exp -4a
13.5 II ( l<) (\ax2 ) I ky2 /s v- 2 (v-l<) (4a ) 1/3 2
Re v > -1
00 k f (x) g(y) = J f(x) (xy) 2J (xy)dx
0 \!
k -ax2 ~(a2-b2)-~y~exp[_~ay2(a2_b2)-11 x·e I (bx 2 ) . 13.6 ~\!
Re \! > -1, a > b . I~V[~by2(a2-b2)-11
x~+2~-\!exp(-~aX2). 'IT-~ r(~+~) 2 3/2 ~ -~\!+ 314 a -~-~~ +~\! r(~-~+\!)
13.7 . I (~ax2) -~-1 y2 y2 ~ Y exp (- 4a)M~+3/2~_~\! ,_~_~~+~\! (2a)
Re \! > 2 Re ~+~ > -~
x~-2~+\!exp(-~ax2) . _~ 3/4+~\!_ 3j2 ~ _~_~\!+~~ ~ -1 (y 2 ) 'IT 2 a y exp - 4a .
13.8 . I~(~aX2) 2 . W 3 (L)
-l<Re \!<~+2 Re ~ ~+~\!- 12 ~ , -~+~~-~\! 2a
X~K (kax2 ) -1 k y2 y2
'ITa y2[I (-) - t.~\! (a:-) 1 ~\! 4 ~\! a 13.9
Re \! > -1
% 2'ITa-2y 312 [I k ~ (y2) y2 X K~\!+~(~ax2) .v- a - '[,~\!_~ (a:-) 1
13.10 Re \! > -1
X 1/3 (\!+~) exp (-l,(ax2) _113 (\!+2) 1/3(V+~) (y2)I (y2) 'ITa y exp - 4a 1/3 (\!+~) 4a
13.11 Kl13 (\!+~) (ax 2)
1.13 Modified Bessel Functions of Other Arguments 119
00 l< f (x) g(y) = f f (x) (xy) 2 J (xy) dx
0 v
X 1/3 (V+la) exp (~ax2) . _1/3 (v+2) 1/3 (~+V) (:C.)K (:C.) a y exp 4a 1/3 (v+la) 4a
13.12 . KI ( +l<) (l;;ax2) /3 v 2
-1 < Re v < 5/2
211+v+l< 'JT~ r (1+2l!+v) x 'exp(-!,-,ax 2) r (Il+V+3j2) 2 % 11 +lav+ 3/4 a -l;;-loll-"V .
13.13 K (~ax2) -1-11 y2 y2) 11 . Y exp (-4a)M!,-,+o/211+lav,!'-'+"Il+"v(2a Re v > -l,Re(211+v»-1
x211+V+laexp(~ax2) 'II" r(l+v+211) 'l4+o/211+lav -!'-'-lall-lav -11-1 r(la-ll) 2 a y .
K (~ax2) y2 y2 11
13.14 . exp (4a) W -l.o- 3;z ll-lav ,~+"ll+lav (2ii) Re v > -I,
-1 < Re (211+V) < 1
X"1 .. v(l.oaX2)Klav(l.oaX2) -1 la y2 y2
a y 1l.ov (4a)Kl.ov (4a) 13.15
Re v > -1
.. (l< 2) -3;z r(la+!,-,v-l.oll)w (:C.)M (y2) x 1l< ( ) ~ax . 2y r (l+!iv) !,-,ll,!,-,V 2a -!,-,ll,!,-,V 2a 4 V-ll
13.16 . Kl;o (V+ll) (!,-,ax 2)
f(x} (xy~J (xy)dx f (x) g(y) = f 0 v
_ 5'2 -1 .rr x K V(ax } i a -ly~[e~irrvK2v (2e ].4a\~) -
13.17 _ 5/2 < Re v < 5/2 ·rr . -].-
_ e-~].rrvK (2e 4a\~) 2v
-2v -1 x Kv_~(ax }
l< -v v-l< l< l< (2rra) 2a y 2J 2v_1 [2ay} 2]K2v_1 [(2ay) 2]
13.18 Re v > 1/6
-2v-2 -1 (2rr) ~a -v-\ v+~K2V [ (2ay)~] J 2v [ (2ay)~] x K 1 (ax ) 13.19 v-~
-~ < Re v < 2
(2ax~) -!.lpra _3/2 a 2 a 2
K2v+1 sec(rrv}y [ll l(-}-Y 1(-)] -v- y -v- Y 13.20
Re v > -1
X-~K (2ax~) -l< a 2 a 2 >.,rrsec (rrv) y 2 [II (-) - Y (-)]
2v -v Y -v Y 13.21
Re v > -~
~y e y
13.22 v v
Re v > -1
f (x)
Re v > - !:!
13.25 !:! K2v+ l (2ax)
13.26 l<
Re v > - !:!
g(y) f (x) (Xy)!:!J (xy)dx v
-v -l< -2v 2v-l a 2 2 Tf "yaK '(-2) V-" y
v -!:! 2v+1 -2v-2 a 2 2 Tf a y K, (2-)
- !:!TfY -!:! Y (~) v Y
13.30
13.32
o
-2 ~ a 2 a 2 -!.ia y Wl< L (2-) W L L (2-) .v , -.l.l Y --.v , -.l.l Y
!.ia-ly~r(~+~l.l+~v)r(~-~l.l+~v) •
. -1 . -1 • W_~V'~l.l(~ay )W_~V'~l.l(-~ay )
-1 -1 • IJ[v+~ (ay ) - yv+~ (ay ) 1
1.13 Modified Bessel Functions of Other Arguments
f (x)
13.35
g(y) = 7 f(x) (XY)~Jv(XY)dX o
-~ v+l-~ v+~ 2 2 ~~-~v-~ a b y (a +y ) •
• K [b(a2+y2)~] ~-v-l
~b~+V+l v+~( 2 2)-~~-~v-~ a y a +y •
123
co
Hv_~(ax) (~~)-~av-~~-V(a2_y2)-~ y < a
14.1 -312 < Re v < % 0 y > a
I<-v ~ -~21-2vr (1-v) [r (~+v)] -1y-~ x' ~ (ax) -Yv (ax) ] . 14.2 -2 < Re v < 2 . (y2_a2)~V-~v-1(~) y > a v a
0 y < a
x V-J.1-~ [U (ax) -Y (ax)] -~2V-J.1 J.1 r(~+v)r(~+v-J.1) -v-~ . ~ a r(~+J.1)r(1+v-J.1) y
14.3 J.1 J.1
-~ Re v<3h,Re(v-J.1»-1 -2 . 2Fl(~+v,~;1+v-J.1;1-a2y )
x V-J.1+~ ~ (ax) -YJ.1 (ax) ] ~ -121+v-J.1a -J.1 3 2J.1-V- 3/2 (v-J.1+1)r(~+J.1) r( h+v)y .
14.4 -2 . 2Fl (1+v-J.1,~-J.1;2+V-J.1;1-a2y )
I< x 2 [II_v (ax) -Y -v (ax) ]
-v -1 v-~-1 2a ~ cos(~v)y (y+a)
14.5 -~ < Re v
2~-1COS(~J.1)r(2+2v+2J.1)aJ.1y-J.1-~ •
14.6 -l<v-~J.1-~ -v-J.1-1 (~) -1 Re(v+J.1)<~-Re J.1 . (a2_y2) 2 P -v-J.1-1 Y
1.14 Functions Related to Bessel Functions 125
00 k
f (x) g(y) = f f (x) (xy) 2J (xy) dx 0 v
Iv_~(ax)-Lv_~(ax) (~n)-~aV-~~-V(a2+y2)-~ 14.7
-~ < Re v < ~
X~[I (ax)-L (ax)] v v 2 -1 v+l -v-~( 2 2)-1 nay a +y
14.8 -1 < Re v < - ~
x~-v+~[I (ax)-L (ax)] n-~2~-V+la~-lyV-2~-~[r(v_~+~)]-1. 14.9 ~ ~
-1 < 2Re ~+Re -2
~+1 < v · 2Fl (l,~;v-~+~;-y2a )
xV-~+~[I (ax)-L (ax)] ~ ~
n -1 2 v-~+ lr (3/av ) [r (3/a~) ]-la~+ly -v-5/2.
14.10 a 2 -1 < Re v < - ~ · 2F 1 (I, 3/2+V; %+~; --)
y2
xV-~-~[I (ax)-L (ax)] n-~2v-~r(~+v) [r(l+~)]-la~y-V-~ . ~ ~
14.11 -2 -~ < Re v< ~ 2Fl(~+v,l:;;1+~;-a2y )
k -1 I-v v-~ -1 x 2[I (ax)-L (ax)] 2n a cos(nv)y 2(a 2+y2) v -v 14.12
Re v > ~
~ -~
126
-2 < Re v < %
-1 - y-v-1 (ax )]
_3~ 2+v-~ 1-~ 'IT 2 cos('IT~)r(3/2+v-~)a •
211-v- 5/2 -2 Y ... 2F 1 (312-~+v,1; 3/2;-a 2y )
~+v -1 ~ 2 r(Yz+~+v)[r(l+~)r(Yz-~)] a •
-1 l< y2 -Yza y 2y (_) Yzv 4a
-1 -l< l< 4 'IT cos ('lTv)y 2K2v [2 (ay) 2]
-1 -l< l< -4'IT a 2COS ('lTv)K_ 2v _1 [2 (ay) 2]
1.14 Functions Related to Bessel Functions 127
'" f l<
f (x) g(y) = fix) (xy) 2J v (xy)dx 0
2v -1 - 3fz 2 - % ~+v . ( ) -v-~ x [~+~ (ax ) - -~ a s~n ~v y
14.20 -1 -yv+~ (ax )] K2V+1[(2aiY)~]K[(-2aiY)~]
-1 < Re v < - 1{6
2v -1 -1 -~ % ~+v -v-~ X [Iv+~ (ax ) -Lv+~ (ax) )] ~ 2 a y .
14.21 J2V+1[(2aY)~]K2V+1[(2ay)~] -1 < Re v < ~ ·
x~(1-fl-V)S fl,V
p~(V+fl-2) · ( :i.) , y > a ~(V+fl) a
0 , y < a
x~-V-flS (ax) -2v-fl ~ r(1-fl-V) -v v-~ fl,-2v-fl 2 ~ r(vv) a y
14.23 Re v>-~,Re(v+fl) < 1 ( 2_a2)~(V+fl-1)pV+fl-1 (:i.) · y V+fl a
xV+f3+~s 13 (ax) 2a+ f3 +va f3 [(v+f3+1)r(1-a-f3)]-1 a, 2
Re v>-1,-1<Re(v+f3)<~-Re 13 · r (3+a+f3+ 2v) -v-2f3- 3/2 . 14.24 2 Y
Re (2v+a+f3) > -3
• 2F 1 (1+f3+v, 1-a+f3· v+ f3 +2 · 1 - a 2 -) 2' , y2
00 k f (x) g(y) = f f (x) (xy) 2JV (xy) dx
0
xv-a-"S 6 (ax) 2V- 1a 6 [f (l+v-a)]-lf (1-a-S+2v) . a, 2
-1 < Re v < 3/2 f(1-a+6+2v) ya-S-v-" 14.25
. 2
Re(2v-a±S) > -1
1-a+6+2v 1-a+6 1-a+v; 1 - a 2 • F
2 , --2-; -) 2 1 y2
-].J-k x 2SV+].J,].J_v+1(ax) 2V-1av-].J-1f(v)y"-V(a2_y2)].J y < a
14.26 Re ].J>-1, -l<Re V< 3/2 0 Y > a
k x 2 [Jv (ax) -Jv (ax)] -1 -k-1 IT sin(ITv)y 2(a+y)
14.27 Re v > -1
k+v v+ 1 v+" 2 2 x 2 U 1(2a2b,ax) (2b) y cos[b(a -y )] y < a v+ 14.28 0 Y > a Re v > -1
,,+v 2 x Uv+2 (a b,ax) (2b)v+1y v+"sin[b(a 2_y 2)] y < a
14.29 0 Y > a Re v > -1
1.15 Parabolic Cylinder Functions
Re v > - lo
v+1< x 2exp (->.,a 2X 2) . 15.2 . D2v+l (ax)
Re v > -1
15.3 D2v_1 (ax)
Re v > - lo
g (Yl = If (xl Cxyl \r (xy) dx o v
2 -2 Lv+lo (loy a )
-2v v-I< -2 a y 'exp (_>.,y 2a ) •
15.4 '{[1-2cos(~v)lD2V_l (ax)- • {[1-2cos(~v)lD2v_l (ya- 1 ) _
- D (-ax) } 2v-l _ D (-ya -1) } 2v-l
Rev>-lo
15.5 '{[1+2cos(~v)lD2V_l (ax)- • {[1+2cos(~v)lD2v_l (ya-1 ) _
- D2v- 1 (-ax)}
v
- D2v+l (-ax) }
• {[1-cOS(2~V)lD2V+1(ax)- '{[1-2COS(~V)D2v(ya-1)+D2v(-ya-l)}
- D2v+1 (-ax) }
• D_ 2v (ax)
Rev>- ~
Re v > - ~
Re v > - ~
-~<Re v<Re(~-211)
Re v > -1
-1 • 0_2v_l(ya )
.yll-~exp(_~y2a-2) •
-1l-1 -2 2 -2 • Y exp(~y2a)W Q(~ a ) a,,,
-2v-l -2 ~sec(wv)a exp(-~y2a) •
-1 -1 • [02v+1 (ay ) - 02v+1 (-ay )]
131
132
15.16
15.17
-1 {[1+2cos(~V)]D2V+1 (ya )-
-1 - D2v+1 (-ya )}
-1 • {[1-2cos(rrv)]D2V+ 1 (ya )-
-1 - D2v+l(-ya )}
-2v-2 v+~ 2 -2 a y exp(-~y a ) •
-1 .{[1-2coS(~V)]D2V+2(ya ) +
-1 + D2V+2 (-ya )}
+ D2V+2 (-ax)}
Re v > -1
-2v-2 v+~ -2 -a y exp(-l,oy2a ) •
-1 • {[1+2cos(TIV)]D2v+2 (ya ) +
-1 + D2v+2 (-ya )}
-1 -v- 3/2 TI sin(TIv)f(2v+3)y -2 -2 exp(l,oy 2a )Kv+1 (l,oy2a )
(2v+1)a-2V-1yV-~exp(l,oy2a-2) •
-1 • D_ 2v_2 (ya )
-2v-2 v+~ -2 -1 a y 2exp (l,oy 2a ) D_ 2v- 3 (ya )
133
134
f(x)
15.26 • D2lJ (ax)
15.27. • DlJ (ax)
o v
2 -2 i -2 • exp(~ a )Wa,B(~ a )
2a = lJ-V-l 2B = lJ+v+l
1I~r(2a) [r(l+v)r(~+B)]-12-B-3/2Va-2a •
V+~ 2 -2 • Y 2F 2 (a,a+~;v+l, B+~;-~ a )
2B = A-lJ+%
135
136
Re ll>-~' Re(411-v»-~
I-v ~-v v-~ -1 2 (~+v)a y Erfc(~a )
1 2 -2 2 -2 'exp(- 'lay a ) M3 L (~y a ) ll-V+.,.,ll
(2a) 211-2vr (211+l) [r (~+v)] -1 •
• yV-~exp (_1/ay 2a- 2 ) D (2-~a-ly) 2v-411
2-~(2a)211-2v-lr{211+1) [r{o/2+v )]-1 •
v+~ 2 -2 -~ -1 'y exp{-Yaya )D2v-411+1{2 a y)
1.16 Whittaker Functions
Re (v-k+1) > 0
Re(v+2k+1) > 0
Re v>-1,Re(v-8k-%)<0
Re (v+8k-%) < 0
= f f (x) (xy) 2JV (xy)dx o
1/3v_2/3_4/3k I/3V- 1/3k- 1/6 'V3k- 1/3v+ 1/ 6 • 2 a y •
• exp(-~y2a-1)I k (l/ay 2a-1 ) 2fa + 1/3 V-l/6
~ - 1/3 v- 2{3 - 4/3 k - 1{3 v- 1/3 k-1j6 4/3 k+ 1/3 v+ 1/6 • 11 2 a y
I 2 -1 1/ 2-1 ·exp (- lay a )I Ihv- 2/ 3 k+ 1/6 ( ay a )
1 2 -1 II 2-1 ·exp(- lay a )K%k_ l/aV_ 1/6 (fay a )
138
16.12 • M. (ax 2 ) --k,ll
Rell>-l:>
Re(211-2k-v) < l:>
Re v > - l:>
Re(v- 2lJ-2k ) < l:>
g (y) '" = f f(x) (xy) 2Jv(xy)dx o
211-V 411-2v v-211-1o 1 2-2 2 a y exp ( lay a ) •
• W (ky 2 a- 2 ) 311-V-1o,1l "
-1 r (211+1) [r (v+k-lJ+l:» ] •
r (2lJ+1) [r (lJ+k+l:» ]-1 .
l:>-ll-k l:>lJ-lok-l:>v+~ lJ+k - 1 • 2 a y.
• exp(_1/ay 2a- 1 )W (~y2a-l) a,13
2a = k+v-3lJ+1o, 213 = k+lJ-v-l:>
• 2lJ-k+loa-l:>k-l:>lJ-l:>v+~yk-lJ-l •
• exp(_1/ay 2a- 1 )M 13(~y2a-l) a,
2a = 3lJ+v+k+l:>, 213 = lJ+v-k+l:>
1 . 16 Whi ttilker Functions
f (x)
Re(k+Il+~v) < l;;
X-~K (~ax2) . ~V-Il
Re v > -J
->;; x M_>;;Il,>;;v(ax)
16.18 . W>;;Il,>;;v(ax)
g (y) k
= f f(x) (xy) 2Jv (xy)dx o
~+k+1l l;;+~k-~Il-~v -1l-k - 1 • 2 a y •
• expe/ay 2a - 1 )W Q (l;;y 2a-1 ) a,,,,
r(I+Il+p+~v)r(I-Il+p+~V)
• [r(1+v)r(3/2+~V+P-k)]-1 •
A=I+p+~v
· M (>;;y 2a-1 ) >;;k+>;;Il,>;;k+l;;v
-1 -Il- k ar(1+v) [r(>;;+>;;v->;;Il)] y •
.
.
ll, 2 V
• Mll,~+v(-iax)
2f (l+2k+ll+~V) [f (1+2ll) ]-1 •
• y-~K, (J..,y 2 a - 1). ,,>V-ll
2 -1 • Mll+2k'll(~y a )
-1 af(l+v) [B(~+~V+ll,~+~V-ll)] •
y < a
o y > a
-~-V ~-v -~-V 2-2 • 2 a P_~+ll(2a x-I) x < a
o x > a
1.16 Whittaker Functions
Re v > -1
Re (v-4y) > -2
YaV,ll
g (y) J"
r(2p)r(p+~)r(p-~) •
-v-l 1-2p v+~ • 2 a y 4F3(P,~+P'P+~'
-1 ~ r (l+~v) [r (~-~+J"v) 1 (2ay) •
141
142
f(x)
2 v, Jl
- 2 v, Jl
Re(v±2Jl) > -1
-1 'W (-iax)
f (x) l.
-1 -l. [r (~+~v+Jl) r (l;;-Jl+l;;v) ] 4ay 2 •
ar(1+v) [r(l;;+l;;v-Jl)]-1y-2Jl-~ •
-l. l. 2Jl l. • (a 2+y2) 2 [a+ (a 2+y2) 2] exp [-b (a 2+y2)2]
1.17 Gauss' Hypergeometric Functions 143
1.17 Gauss' Hypergeometric Function
17.1 Re v<-~, Re ).,>0 -i1f -1 i1f -L
• [W~_a., -~-v (ye )., ) -W~_a. , -~-v (ye )., -)]
Re (a.+v) > - ~
-1 < Re v < 17.3
Re )., > 0
1f -~ [r (a.) r «(3)(121-(3)., -v-(3-1 r (l+~v+~(3) •
-1 2 • [K~(v_(3+l) (~)., )]
f(x) (XY~Jv(XY)dX f(x) g(y) = f 0
-201- 3/2 x 2Fl(~+a,1+a1 A-2aY~I~v+a(AY)K~v_a(AY)
17.5 1+2a1-4A 2X-2)
x ~+v-4a 2 VA 1-2ar (v) [r (201) ]-ly2a-V-~ •
2 -2 '2Fl (a,a+~1v+11-A x ) . Iv(~AY)K2a_v_1(~AY)
17.6 -1+Rea<Rev<4Rea- 3k
Re A > 0
xV+~(l+x)-2a • [r(a)]-l r (V+1)r(1+v_a)22v-2a+1 . 17.7 -2
'2Fl[a,~+v12v+114x(1+x) ] 2a-2v- 3/2 • y Jv(Y)
-1 < Re v<2 Re a-~2
Chapter II. Integral Transforms with Modified Bessel Functions as Kernel
A representation of a given function f(x) by means of a double integral involving modified Bessel functions of order v is
f(x) c+i~ ~
(d) -1 f I (tx) (tx) Ijdt f K (ut) (ut) lzf (u) du c-i~ v 0 v
or also
c+i~ ~
f (x) = (2d) -1 f [I (tx) +1 (tx) ] (tx) Ijdt f K (ut) (ut) Ijf (u) du c-i~ v -v 0 v
This is equivalent with the pair of inversion formulas
(1) g(y;v) j f(x) (xy)lzK (xy)dx o v
(2) f(x) c+ioo
or
(3) g(y;v) j f (x) (xy) lzK (xy) dx o v
(4) f(x) = (2d)-1 c+ioo f g(y;v) (xy)lz[1 (xy)+I (xy)]dy
v -v c-ioo
1lz(z) = (lzuz)-lzsinhz, 1_lz(Z) = (ljuz)-~cosh z
the equations (3) and (4) become
146
-1 l.: (21T ) 2g (Y; ±l;,) f f (x)e -xy dx o
fix) = (21Ti)-1 f (21T- l ) g(y;±l;,)exy dy o
These are the Laplace transform formulas.
Since
II. K-Transforms
it is possible to evaluate integral transforms with Bessel func tion (Chapter I) or Neumann function kernel (Chapter III) by means of the above relations.
REFERENCES
Boas, R. P., 1942: Proc. Nat. Acad. Sci. U.S.A. 28, 21-24.
Boas, R. P., 1942: Bull. Amer. Math. Soc. 48, 286-294.
Meijer, C. S., 1940: Proc. Amsterdam Akad. Wet. 702-711.
599-608;
0 v
1 c+i ... g(y)(xy)~ 2d .
. 1.2 c-], ... g(y) . [Iv(xy)+I_v(XY)]dy
1.3 f(ax) -1 -1 , a > 0 a g(ya iV)
xmf(x), m = 0,1,2,'" ~-v d m [ym+v-~g (y i m+v) ] 1.4 y (- ydy)
~+v(_ ~) m
[ym-v-~g (y i v-m) ] 1.5 xmf(x), m = 0,1,2,'" Y ydy
1.6 -1 x f(x) -1 ~v [yg(Yiv+l)-yg(YiV-l)]
x-).lf(x) 21-).l[r().l)]-lyv+~ j T~-).l-v(T2_y2»).l-1.
1.7 Re ).l > 0 y
• g(TiV+).l)dT
1.8 f' (x) -1 ~v [ (v-~) yg (y i v+l) + (v+~) yg (y iv-I)]
148 II. K-Transforms
2.2 Transforms of Order Zero
00 k f (x) g (y) = J f (x) (xy) 2KO (xy) dx
0
2.1 xy,(a 2+X2)-Y, -k 2y 2[sin(ay)Ci(ay)-cos(ay)si(ay)]
x-Y,(a2+x2) -Y, · ~na-2~yY,~[J +k (y,ay)Y k (y,ay) -v ~ 2V . ~- 2V 2.2
[x+(a2+x2)Y,]-2~ . - Y +' (y,ay)J k (y,ay)] -0 ~ ~v ~-2V v-
~ ~
+ [ (a2+x2)Y,_x]2~}
0 x < a
2.4 xy,-V(x 2_a 2)V-y, x>a nY,2v- 1 (y,+v)y -v-Y,e -ay
Re v > - Y,
0 x < a
Y,-v 2 2 v- 3/2 Y, v-2 -1 Y,-v -ay 2.5 x (x -a ) x>a n 2 a r(v-y,)y e
Re v > Y,
0 x < a
x-Y,(x 2_a 2)-Y, · 2~+1 Y,[K (k )]2 a y ~+Y, 2 ay
2.6 .{[2x2-a2+2x(x2_a2)Y,]~+Y,+
co
X~(b2+X2)-~ y-~[sin(bY)Ci(by) - 2.7
- cos(by)si(by)]
x-~e-ax ~ 2 2 -~ -1 2 -2 ~ y'(a-y) 1og[ay +(ay _1)2] y<a 2.8
~ -~ -1 Y 2 (y2_a 2) "arccos (ay ) y>a
X~(b2+X2)-~ . ~ -~ ~y (a2_y2) 2 . ~ {eXp[b(a2-y2)~]Ei(-Zl) . exp[-a(b2+x2) 2] . -
2.9 ~ . - exp[-b(a 2_y 2) 2]E~(-z )}
2
1 2
-~ ~ [(-2iay) ~] 2.10 2ay K [(2iay) 2]K 1 1
_3/2 -1 2Y~Ko[(2iaY)~]Ko[(-2iay)~] 2.11 -ax x e
2.12 x-~sin(ax) ~ -~ -1 2 -2 ~ y2(a 2+y2) 21og[ay +(1+a y ).2]
2.13 x-~cos(ax) ~ -~ ~1TY (a 2+y2) 2
co k f (x) g(y) = f f (x) (xy) 2KO (xy) dx
0
x-\log(ax)cos(bx) \7TY\(b2+y2)-~ . 2.14 . [log(\ay)-y-log(b 2+y2)]
x\sin (ax-I) -k [(2aY)\]K
k 2.15 7Tay 2 [J [ (2ay) 2] 1 1
2.16 x\cos(ax-1 ) -k ~. \ -7Tay 2[y [(2ay) ]K [(2ay) ] 1 1
_ 3/2 -1 k k ~ 2.17 x sin (ax ) 7Ty2JO[(2ay) 2]K O[(2ay) ]
2.18 -312
cos (ax-I) -7TY~Yo[(2aY)~]Ko[(2ay)~] x
X~(b2+X2)-~ . \y\(a 2+y2)-\{sin u[Ci(z )+Ci(z ) ] - 1 2
. cos[a(b2+x2)~] - cos u lsi (z ) + si(z )]} 2.19 1 2
k U = b(a 2+y2) 2
z = b[(a2+y2)~±a] 1 2
k -k k _\ x 2 (b2+x2) 2 . ~y2(a2+y2) {cos u[Ci(z )-Ci(z ) ] + 1 2
2.20 'sin [a (b2+X2)~] + sin u[si(z ) - si(z ) ] } 1 2
U, Z 1 '
2.21
2.22
2.23
lzY~(a2-y2) -~ •
- sin u[Ci(z ) + Ci(z )]} y < a 1 2
u = b(a2_y2)~
• y {~a[b+(b2_y2)~]} + o
+ J {~a[b+(b2_y2)~]} • o
1 S1
'" l< f (x) g(y) = f fix) (xy) 2KO (xy)dx 0
2.26 _ 3/2
x-~(a2-x2)-~ l< l< l<
h[b(a2-x2)~]x<a ~lIY 2[IO {~a [(b 2+y2) 2_b] }KO {~a [(b 2+y2) 2+b]}+
2.27 cos
0 x>a + IO{~a[ (b2+y2)~+b] }KO{~a[ (b2+y2)~_b]}]
-l< l< -l< -l< 2.28 x 2J O (ax) y 2 (a 2 +y 2 ) "K [ a (a 2 ,y 2 ) 2]
2.29 x-~yo(ax) l< 2 2 -l< 2 2-l< - y2(a +y ) 'a[y(a +y ) 2]
-l< -l< -1 2.30 x 2IO (ax) y>a y "I«ay )
2.31 IO(ax) y>a 2~TI-1r 2 (3/4 ) (y2_a2)-~ . -2 l< l<
• IC{ [~-~ (1-a 2y ) 2] 2}
-l< -1 l< 2 -2 l< x 2KO (ax) ~a y "Ic[ (1-y a ) 2] Y < a
2.32 _l< 2 - 2 l<
~y "1([ (a y -1) 2] Y > a
f (x) g (y) = 7 f(X)(XY)~O(XY)dX 0
-1 2- 312r 2 (~) (:i.) ~ • x KO (ax) a
~{[~_~(1_y2a-2)~]~} +
(~)~K{[~_~(1_a2y-2)~]~} + y
x 'KO (ax) ~ -1 -1 y (a 2_y2) 1og(ay )
-k x 'cos (bx) 10 (ax) k -~ k -k y'[b 2+(a+y)2] {2(ay)2[b2+(a+y)2] 2}
2.35 Y ;:, a
-k ~ b2 ( 2 ." 2.36 x 2COS (bx)KO (ax) 1IY [b 2+ (a+y) 2 ]K{ [ + a-y) ] }
b2+(a+y)2
_k 2 -1 ." k -1 x 2[JO (ax)] 211 y [2a+ (4a 2+y2) 2] • 2.37
k ." -1 • K2{2a'[2a+(4a 2+y2)] }
-k 2 x 2[1 O(ax)] 211-1y-~2{["'_"'(1_4a2y-2)"']"'} 2.38
y > 2a
f(x) g(y) = j f(x) (XY)~KO(XY)dX 0
-~ 2 ~~a-1y~{[~_~(1_~2a-2)~]~} x [KO (ax)] . 2.39 • • {[~+~(1_~2a-2)~]~} y < 2a
0 y > 2a
2.40 X-~Io(aX)Ko(aX) Y-~{[~-~(1_4a2y-2)~]~} •
• K{[~+~(1-4a2y-2)~]~} y > 2a
312 312 -2 -1 -~ -1ac-1 x Io(ax) 2y (y2_a 2) Stay )-y (y2_a 2) (ay) 2.41
y < a
2.42 x~Jo(aX)Jo(bX) y~[(a2+b2+y2)2-4a2b2]-~
2.43 X~[Jo(~ax2)]2 1(1 6~a -ly~{ [J 0 (1/ey 2a -1) ]2+ [YO (lfey 2a -1) ]2}
2.44 X~J (ax~)I (ax~) -% -1 o 0 y cos(~a2y )
4Jo(aX~)Ko(aX~)+ - 3/2 -1 2~y sin(~a2y )
2.45
+2~Io(aX~)Yo(ax~)
2.46 X-~[K (~ax-1)]2 2~y-~K2~[(2iaY)~]K2~[(-2iay)~] ~
CD
f(x) (XY)~KO(XY)dX f(x) g(y) = J 0
2.47 x-~J (~ax-1)y (~ax-1) jJ jJ -2y -~J 2jJ [(2ay) ~]K2jJ [(2ay)~]
x-~{[J (~ax-1)]2 - jJ 4Y-~2jJ[(2aY)~]K2jJ[(2ay)~]
2.48 _[YjJ(~aX-1)]2}
2.49 .~ (-iax2 ) IjJ .w k(~iy2a-1)W k[-~iy2a-1)
-jJ I -jJ I
RejJ>-~
-% 1/16'11"Y~{ [Jk ('iey2a -1)] 2+ [Yk (1;ay2a -1)] 2} x ~,0(iax2) . 2.50
.M (-iax 2 ) -k,O
0 x < a
x-~(a+x)-1
-Yo-A -Yo~iA . ay e [Kv_l(ay)SA+l,v(~ay)
+ i(V+A)K (ay)S, l(iay)] v /\,v-
F (1; 3/2-~~-~V, 3/2-Yo~+~v;"'a2y2) - 1 2
~ k -~a y2cSC[~(~-v)] •
2.3 Elementary Functions 157
x-~(a2+x2)-~ 1/81T2sec(~1TV)Y~ • 3.6
-1 < Re v < 1 '{[J~V(~ay)]2 + [Y~v (~ay) ]2}
x-~(a2+x2)-1 ~1T2a-1sec(~1TV)Y~ •
3.7 -1 < Re v < 1 · {tan(~1Tv) [3v (ay)-Jv (ay)] -
- BV(ay) - Yv(ay)}
Re v < ~ · [HV(ay) - YV(ay)]
Re (ll±V) < ~ • y~s (ay) ll,V
x~+V(a2+x2)ll 2vr(1+v)aV+ll+1y-ll-~ • 3.10
Re v > -1 · Sll-V,ll+V+1(ay )
158
3.11
f(x)
3 • 12. • [ (a 2 +x 2 ) ~_ a ] - k •
3.13
3.14
o x>a
• F (-~;1-~-~A-~v/1-~-~A+~V;-~a2y2) 1 2
• yV F (~A+~V;~A+~+lt~v/1+v;-~a2y2) 1 2
2.3 Elementary Functions
o
o v
,,-1 "-v+1 -"-k +v2~ a~ y ~ 2CSC(VV) •
• r(~+1)I +1(ay) ~-v
~-v+1
[ (a2+x2) !o_x]2j.l, }
{C05[!orr(v-j.l)] [(a 2+x2)!o+x]2j.l+
+C05 [!orr (V+j.l)] [(a 2+X2) !o_x]2j.l}
Re (2j.l±v) < 1
v
- Y +1. (!oay)J I. (!oay)] j.l 2V j.l- 2V
• W I. (iay)W I. (-iay) j.l, 2V j.l, 2V
2.3 Elementary Functions
principal value
3.27 Rev>-~
principal value
-1 -J. TIa y 2W J. (ay)W J. (ay)
1l,2V -1l,2V
1 -1 - IBTI 2 a sec (~TIv) [I (ay) +1 (ay) 1 } v -v
.{cot[~TI(ll-v)l1_v(ay) +
Re(p±v) > -1
3.32
f(x) (XY)~K (xy)dx g (y) = f v 0
~~csc(TIv)y~-V(a2-y2)-~ . k V k V • {[a+ (a 2_y2) 2] - [a- (a 2_y 2)'J }
k -k TICSC (~v)y 2(y2_a 2) 2 . -1 · sin[varccos(ay )J
k -I< TICSC (TIv)y 2(y2_a 2) 2 . · sin{v[~~+arcsin(ay-1)J}
k (~TI) 'f (p+l-v)f (p+l+v) •
• (y2_a2)-~p-~p-p-~(ay-l) = v-~
= f(p_v+l)y~(a2_y2)-~p-~e-iTIv
k k k k 2y2K {a 2 [(b+y) 2+ (b-y) 'J} •
v
3.34
3.35
3.36
3.37
-1 < Re v < 1
~ sec(~'II"v) (~)~ • a
x..: ~ • exp(Sa)WjJ,~v(4a)
~y~sec (~'II"v) •
Sa '" 4
-~ < Re v < ~
3.40 x < a
o x > a
y2 11 11 (8a - 4" - 4" v) -
y2 11 11 (8a - 4" - 4" v)]
h -h-h (~11) 2r (~-v)D L[a(-iy) ']D ,[a(iy) 2]
V-'2 V-'2
165
2 k ~8 ec(l:i~v)y2[YL (U)Jk (V)-JL (U)Yk (v)]
~V 2V ~V 2V
sin [varcsin (~) ] y
cos [varcsin (~) ] y
00 k f (x) g(y) = f f (x) (xy) 2Kv (xy) dx
0
_ 3/2 sin h(ax) -k -1 x Y,7TY 2V sec (Y,7TV) .
3.47 -1 < Re v < 1 sin [varcsin (~) ] .
-Y, 2 2-~ X ta -x ) . k ~7T2Y'CSC(Y,7TV) [I_y,v(U)I_y,v(V) -
3.48 ·cos h[b(a 2-x 2)l:i] - Il:iv(U)I~v(V)]
x < a ~a [(b 2+y2) ~+b]
0 U = x > a
-1 < Re v < 1 ~a[ (b2+y2)~_b)] v =
0 x < a -1 -l:i l:i7Ta y Wk k (ay)W k l:i (ay) .].1 , .v - .].1, v
x-l:i{x 2_a 2)-l:i 3.49
-1 cos[].Iarccos(ax )]
2.4 Higher Transcendental Functions
o x < a
4.1 x > a
4.4 x > a
Re ]1 < 1
o x < a
'" k f (x) g(y) = f f (x) (xy) 2KV (xy) dx
0
x~-1(x2_a2)-Jo~p~ (~) ~ U-Jo,v-~
2 ay 2S2VI2~+1 (ay) 4.7
Re v < 1
k: V k:: x 2p [(I+x2)2] ~ -1 4.8 y Sv+Jo,u+Jo(y) Re v < 1
k -Jo v k x 2 (I+x 2) P [(I+x2) 2] ~
4.9 S (y) Re v < 1 V-Jo/~+Jo
~
f(x) g (y) k
x~(a2+x2)':;V-l
2 y S2v_l,2jl(ay ) 4.11
+pv (l+2x2a-2)] -jl
-v v-k 2 ay 2Kll+1 (ay)
x > a
Re v < 1
0 x < a
(x 2_a 2) ,:;v->" . 4.13 k-v -2 .p" (2x 2a -1) jl
-k v-I ~-v 2 'IT "2 ay [K +k (~ay)] jl 2
x > a
Re v > - ':;
x-V-~(a2+x2)>"-':;V . . -i'ITV ~2 -v-3 ,:;-v v-':; 2 ~e 'IT 2 a y [r(l-v)] •
4.14 k-v -2 .q.\ (l+2a 2x ) {[Jv_~(,:;ay)]2 + [Yv_~(,:;ay)]2}
Re v < 1
'" I< f (x) g(y) = f f(x) (xy) 'KV (xy) dx
0
x-V-~(a2+x2)~-~V ,
,q~-V (l+2a 2X -2) i e-i1TV1T~2-V-la-V-\V-3/2 [r (3/2+)J-V) ]2,
)J , W ~ l«iay)W I< ~(-iay) 4.15 -)J- ,V-2 -)J-.,v- Re )J>_3/2
Re (]ol-v) > - 312
XV+)J+~J (ax) 2v+)Jr(V+)J+l)a)Jyv+~(a2+y2)-)J-v-l 4.16 )J
Re )J>-I, Re(v+)J) >-1
4.17 J (ax)
)J 2)J-~r (%+~)J+~v) r (3/4+~)J-~V) Re()J±V) > - % '(a2+y2)-~-)J [(I+a2y-2)~]
v-~
x)J-~J )J (ax) 2)J-1r(~+)J+~v)r(~+)J-~V) ,
4.18 -I< -I<)J -)J -2 Re (2)J±v) > -1 .y 2(a 2+y2) • p (1+2a 2y) ~v-~
x-)J-~J (ax) -)J-l -I< 1T2 sec(~~v)y 2 . 4.19 )J
-1 < Re v < 1 (a2+y2)~)Jp-)J (l+2a 2y-2) , ~v-~
X-A-~J v(ax) 2-A-1r(~+V-~A)r(~-~A)Y~ . 4.20
Re A<I, Re(2v-A»-1 (a2+y2)-~+~Ap-V (y2_a 2) ~A-~ y2+a 2
2.4 Higher Transcendental Functions
4.21 Re(j.!±v-A) > -1
Re(±j.!±v-A) > -1
-1 < Re v < 1

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