This article was downloaded by: [University of Sydney] On: 13 March 2013, At: 05:45 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Integral Transforms and Special Functions Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gitr20 On the derivatives of the Bessel and Struve functions with respect to the order Yu. A. Brychkov a & K. O. Geddes b a Department of Special Functions, Computing Center of the Russian Academy of Sciences, Vavilov Street 40, Moscow 117967, V-333, Russia b Faculty of Mathematics, School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada Version of record first published: 26 Jan 2007. To cite this article: Yu. A. Brychkov & K. O. Geddes (2005): On the derivatives of the Bessel and Struve functions with respect to the order, Integral Transforms and Special Functions, 16:3, 187-198 To link to this article: http://dx.doi.org/10.1080/10652460410001727572 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and- conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
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This article was downloaded by: [University of Sydney]On: 13 March 2013, At: 05:45Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Integral Transforms and SpecialFunctionsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gitr20
On the derivatives of the Bessel andStruve functions with respect to theorderYu. A. Brychkov a & K. O. Geddes ba Department of Special Functions, Computing Center of theRussian Academy of Sciences, Vavilov Street 40, Moscow 117967,V-333, Russiab Faculty of Mathematics, School of Computer Science, Universityof Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1,CanadaVersion of record first published: 26 Jan 2007.
To cite this article: Yu. A. Brychkov & K. O. Geddes (2005): On the derivatives of the Bessel andStruve functions with respect to the order, Integral Transforms and Special Functions, 16:3, 187-198
To link to this article: http://dx.doi.org/10.1080/10652460410001727572
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions
This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any representationthat the contents will be complete or accurate or up to date. The accuracy of anyinstructions, formulae, and drug doses should be independently verified with primarysources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand, or costs or damages whatsoever or howsoever caused arising directly orindirectly in connection with or arising out of the use of this material.
Integral Transforms and Special FunctionsVol. 16, No. 3, April 2005, 187–198
On the derivatives of the Bessel and Struve functions withrespect to the order
YU. A. BRYCHKOV† and K. O. GEDDES‡*
†Department of Special Functions, Computing Center of the Russian Academy of Sciences,Vavilov Street 40, Moscow 117967, V-333, Russia
‡Faculty of Mathematics, School of Computer Science, University of Waterloo,200 University Avenue West, Waterloo ON N2L 3G1, Canada
(Received 20 May 2003)
Closed form expressions are obtained for the first derivatives with respect to the order of the Besselfunctions Jν(z), Yν(z), Iν(z), Kν(z); integral Bessel functions J iν(z), Y iν(z), Kiν(z); and Struvefunctions Hν(z), Lν(z) at ν = ±n, ν = ±n + 1/2, with n = 0, 1, 2 . . . .
The expressions for some first derivatives of the Bessel and Struve function can be found inrefs. [1, 2]. We present them here as they will be used below in deriving general expressions.
2+ sin z[Si(2z) − 2Si(z)] + cos z[ci(2z) − 2ci(z)]
}, (11)
H∗−1/2(z) =
√2
πz{cos z[Si(2z) − 2Si(z)] − sin z[ci(2z) − 2ci(z)]}, (12)
L∗1/2(z) = 1√
2πz
{−
(C + ln
z
2
)+ 2sinh z[shi(2z)
− 2shi(z)] − 2 cosh(z) [chi(2z) − 2chi(z)]}, (13)
L∗−1/2(z) =
√2
πz{cosh z[shi(2z) − 2shi(z)] − sinh z[chi(2z) − 2chi(z)]}. (14)
Let us note that we use representations (9), (10) and (13), (14), which differ from thosegiven in refs. [1, 2] as they are more convenient in the complex domain. They can be derivedby the method of ref. [2, 7.8], where the derivatives were expressed in terms of Ei(−z) andEi(z). We use the following notations:
Si(z) =∫ z
0
sin x
xdx, ci(z) = −
∫ ∞
z
cos x
xdx,
shi(z) =∫ z
0
sinh x
xdx, chi(z) = −
∫ ∞
z
cosh x
xdx,
C = 0.5772156649 . . . is the Euler constant. The other definitions are standard [see, e.g.,ref. 3]. Moreover, for every function fν(z) we will use the notations
Dn[fν(z)] = ∂nfν(z)
∂zn, f ∗
a (z) = ∂fν(z)
∂ν
∣∣∣∣ν=a
.
2. Bessel functions Jν(z), Yν(z), Iν(z), Kν(z)
To derive the closed expressions for derivatives of the Bessel functions, we will start from thewell-known differentiation formulas
Dn[z±ν/2Jν(√
z)] =(
±1
2
)n
z(−n±ν)/2Jν∓n(√
z), (15)
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Derivatives of Bessel and Struve functions 189
Dn[z±ν/2Yν(√
z)] =(
±1
2
)n
z(−n±ν)/2Yν∓n(√
z), (16)
Dn[z±ν/2Iν(√
z)] = 1
2nz(−n±ν)/2Iν∓n(
√z), (17)
Dn[z±ν/2Kν(√
z)] =(
−1
2
)n
z(−n±ν)/2Kν∓n(√
z), (18)
which are usually written using the operator ∂/z∂z [1]. By dividing, for example, equation (15)by (±1/2)nz(−n±ν)/2, differentiating with respect to ν, changing the order of differentiationsand substituting ν = 1/2, we get the relation
J ∗1/2±n(
√z) = ∓1
2(±2)nz(2n±1)/4 ln zDn[z±1/4J1/2(
√z)]
+ (±2)nz(2n±1)/4Dn[z±1/4 ln zJ1/2(√
z) + z±1/4J ∗1/2(
√z)]. (19)
Now using the formula
J1/2(z) =√
2
πzsin z
and the relations
Dn[sin√
z] =√
π
2n+1/2z(1−2n)/4J1/2−n(
√z), (20)
Dn
[sin
√z√
z
]= (−1)n
√π
2n+1/2z−(1+2n)/4Jn+1/2(
√z), (21)
[which follow from equation (15)] and equation (5), we derive, after some algebra, the closedexpressions for J ∗
In this section, we consider the integral Bessel functions
J iν(z) =∫ ∞
z
1
xJν(x) dx, (31)
Y iν(z) =∫ ∞
z
1
xYν(x) dx, (32)
Kiν(z) =∫ ∞
z
1
xKν(x) dx. (33)
Differentiating the formula (31) with respect to the order ν and using equation (1), we get
J i∗±n(z) = (±1)nπ
2
∫ ∞
z
1
xYn(x) dx ± (±1)n
n!2
n−1∑k=0
2n−k
k!(n − k)
∫ ∞
z
xk−n−1Jk(x) dx,
where n = 0, 1, 2, . . . . For the second integral, we have
∫ ∞
z
xk−n−1Jk(x) dx = −n−k−1∑
r=0
(−1)rzr−n+k
(n − k)r+1Dr [Jk(x)]
+ 1
(n − k)!∫ ∞
z
1
xDn−1[Jk(x)] dx
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Derivatives of Bessel and Struve functions 193
after successive integrations by substitution. Now, from the last two formulas and the relation
Dn[Jν(z)] = 1
2n
n∑p=0
(−1)p(
n
p
)Jν+2p−n(z)
[see, e.g., ref. 3, 4.2.1.10], we obtain the expression
J i∗±n(z) = (±1)nπ
2Y in(x) ± (±1)n
n!2
n−1∑k=0
1
k!(n − k)
× 1
(n − k)!n−k∑p=0
(−1)p(
n − k
p
)J i2p+2k−n(z)
−n−k−1∑m=0
(−1)m2n−k−mzk+m−n
(k − n)m+1
m∑p=0
(−1)p(
m
p
)J i2p+k−m(z)
. (34)
Similarly, using equations (32) and (2) and equations (30) and (4) we get
Y i∗±n(z) = −(±1)nπ
2J in(x) ± (±1)n
n!2
n−1∑k=0
1
k!(n − k)
× 1
(n − k)!n−k∑p=0
(−1)p(
n − k
p
)Y i2p+2k−n(z)
−n−k−1∑m=0
(−1)m2n−k−mzk+m−n
(k − n)m+1
m∑p=0
(−1)p(
m
p
)Y i2p+k−m
, (35)
Ki∗±n(z) = ±n!2
n−1∑k=0
1
k!(n − k)
1
(n − k)!n−k∑p=0
(−1)n−k
(n − k
p
)Ki2p+2k−n(z)
−n−k−1∑m=0
2k−n−mzk+m−n
(k − n)m+1
m∑p=0
(m
p
)Ki2p+k−m(z)
. (36)
Some special cases are
J i∗0 (z) = π
2Y i0(x), Y i∗0 (z) = −π
2J i0(x), Ki∗0 (z) = 0,
J i∗±1(z) = ±π
2Y i1(x) + J i1(x) + 1
zJ0(x),
Y i∗±1(z) = ∓π
2J i1(x) + Y i1(x) + 1
zY0(x),
Ki∗±1(z) = ∓π
2Ki1(x) ± 1
zK0(x).
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194 Yu. A. Brychkov and K. O. Geddes
4. Struve functions Hν(z) and Lν(z)
The following differentiation formulas for the Struve functions Hν(z) and Lν(z) can beobtained for any n = 0, 1, 2, . . . ,
Dn[zν/2Hν(√
z)] = 1
2nz(−n+ν)/2Hν−n(
√z), (37)
Dn[z−ν/2Hν(√
z)] =(
−1
2
)n
z−(n+ν)/2Hν∓n(√
z)
− (−1)n
π
n−1∑k=0
�(k + 1/2)
�(ν + n − k + 1/2)
z−k−1/2
2ν+2n−2k−1, (38)
Dn[zν/2Lν(√
z)] = 1
2nz(−n+ν)/2Lν−n(
√z), (39)
Dn[z−ν/2Lν(√
z)] = 1
2nz−(n+ν)/2Lν∓n(
√z)
+ 1
π
n−1∑k=0
(−1)k�(k + 1/2)
�(ν + n − k + 1/2)
z−k−1/2
2ν+2n−2k−1. (40)
Applying the method of the previous section and making use of the formulas (37)–(40) and(11)–(14), we easily derive the formulas for the first derivatives with respect to ν for thefunctions Hν(z) and Lν(z) at the point ν = ±n − 1/2.
The last formulas we get in this article are the expressions for Hν(z) and Lν(z) for ν =0, ±1, ±2, . . . . Note that the restrictions on z can be removed by analytic continuation. Letus consider the integral [3, 2.13.16.2],∫ ∞
0J2ν(2
√xz)Yν(x) dx = −Hν(z),
where z > 0, Re ν > −1/2. Differentiating with respect to ν and putting ν = 0, we get
π
∫ ∞
0Y0(2
√xz)Y0(x) dx − π
2
∫ ∞
0J0(2
√xz)J0(x) dx = −H∗
ν(z).
The second integral is given in ref. [3, 2.12.34.1]:∫ ∞
0J0(2
√xz)J0(x) dx = J0(z).
The first one can be evaluated by general method using the Mellin transformation [see, e.g.,refs. 5 or 6, 8.1]:
∫ ∞
0Y0(2
√xz)Y0(x) dx = −2π
zG32
46
z2
4
∣∣∣∣∣∣∣∣1, 1,
1
4,
3
41
2, 1, 1,
1
4,
1
2,
3
4
where Gmnpq
(z
∣∣∣∣(ap)
(bq)
)is the Meijer G-function [see ref. 6, 8.1], (ap) = a1, a2, . . . , ap, and
(bq) = b1, b2, . . . , bq .Now, we have the equality
H∗0(z) = 2π
zG32
46
z2
4
∣∣∣∣∣∣∣∣1, 1,
1
4,
3
41
2, 1, 1,
1
4,
1
2,
3
4
+ π
2J0(z). (45)
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196 Yu. A. Brychkov and K. O. Geddes
The combination of the formulas (15), (37), and (38) and differentiation formula for the MeijerG-function
Dkz
[z−b1−1Gmn
pq
(z
∣∣∣∣(ap)
(bq)
)]= (−1)kz−b1−k−b1−1Gmn
pq
(z
∣∣∣∣ (ap)
b1 + k, b2, . . . , bq
),
where m � 1 [6, 8.2.1.38], gives
H∗−n(z) = π
(2
z
)n+1
G3246
z2
4
∣∣∣∣∣∣∣∣1, 1,
1
4,
3
4
n + 1
2, 1, 1,
1
4,
1
2,
3
4
+ (−1)n
π
2Jn(z)
− 1
2
n−1∑k=0
(n
k
)(n − k − 1)!
(− z
2
)k−n
H−k(z), (46)
H∗n(z) = (−1)nπ
(2
z
)n+1
G3246
z2
4
∣∣∣∣∣∣∣∣1, 1,
1
4,
3
4
n + 1
2, 1, 1,
1
4,
1
2,
3
4
+ π
2Jn(z)
+ 1
2
n−1∑k=0
(−1)k(
n
k
)(n − k − 1)!
(− z
2
)k−n
H−k(z)
+ 1
π
n−1∑k=0
�(k + 1/2)
�(n − k)
( z
2
)n−2k−1[
lnz
2− ψ
(n − k + 1
2
)]. (47)
The formula [3, 2.16.22.5],
∫ ∞
0J2ν(2
√xz)Kν(x) dx = π
2Iν(z) − π
2Lν(z)
after differentiation with respect to ν gives for ν = 0
π
∫ ∞
0Y0(2
√xz)K0(x) dx = −π
2K0(z) − π
2L∗
0(z),
where the formulas (1) and (4) were used. Evaluating the integral, we get
L∗0(z) = −K0(z) − 2
zG42
46
z2
4
∣∣∣∣∣∣∣∣1, 1,
1
4,
3
41
2,
1
2, 1, 1,
1
4,
3
4
.
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Derivatives of Bessel and Struve functions 197
Using the formulas (10), (39), and (40), we derive that
L∗−n(z) =
(−2
z
)n+1
G3246
z2
4
∣∣∣∣∣∣∣∣1, 1,
1
4,
3
4
n + 1
2,
1
2, 1, 1,
1
4,
3
4
− (−1)nKn(z)
+ 1
2
n−1∑k=0
(n
k
)(n − k − 1)!
(− z
2
)k−n
L−k(z) (48)
L∗n(z) = π
(−2
z
)n+1
G3246
z2
4
∣∣∣∣∣∣∣∣1, 1,
1
4,
3
4
n + 1
2,
1
2, 1, 1,
1
4,
3
4
− (−1)nKn(z)
+ 1
2
n−1∑k=0
(n
k
)(n − k − 1)!
(− z
2
)k−n
L−k(z)
− 1
π
n−1∑k=0
(−1)k( z
2
)n−2k−1[
lnz
2− ψ
(n − k + 1
2
)]. (49)
The alternative expressions for H∗±n(z), L∗±n(z) can be obtained in the following way. Letus consider the value of the integral [3, 2.16.27.4],∫ ∞
0xYν(x
2)K2ν(2√
zx) dx = π
8[csc νπH−ν(z) − sec νπY−ν(z) − 2csc νπHν(z)],
where z > 0, |Re ν| < 1, at the point ν = 0. Using the L’Hôpital’s rule in the right-hand side,we obtain ∫ ∞
0xY0(x
2)K0(2√
zx) dx = −π
8H∗
0(z) − π
8J0(z).
Evaluating this integral we get the relation
H∗0(z) = 1
πzG32
24
z2
4
∣∣∣∣∣∣∣1, 1
1
2, 1, 1,
1
2
− π
2J0(z). (50)
In order to derive a new expression for L∗0(z), we can use the integral [3, 2.16.33.3],∫ ∞
0Kν(x)K2ν(2
√xz) dx = π
4sec νπ{Kν(z) − π
2νcsc νπY−ν(z)
− 2csc νπ[L−ν(z) − Lν(z)]}with Re ν > 1/2. If we evaluate this integral at ν = 0 and apply the limiting process in theright-hand side, we come to the equality
L∗0(z) = − 1
π2zG42
42
z2
4
∣∣∣∣∣∣∣1, 1
1
2, 1, 1,
1
2
+ K0(z). (51)
Starting from the formulas (50), (51) and using equations (1), (4), (37)–(40), we can derivenew representations for H∗±n(z) and L∗±n(z).
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198 Yu. A. Brychkov and K. O. Geddes
Similar formulas were obtained for the Anger function Jν(z) , Weber function Eν(z), andKelvin functions berν(z), beiν(z), kerν(z), keiν(z).
References[1] Magnus, W., Oberhettinger, F. and Soni, R.P., 1966, Formulas and Theorems for the Special Functions of
Mathematical Physics. 3rd edition (New York: Springer–Verlag).[2] Luke, Y., 1962, Integrals of Bessel Functions (New York: McGraw-Hill).[3] Prudnikov, A.P., Brychkov, Yu.A. and Marichev, O.I., 1986, Special Functions. Vol. 2. Integrals and Series
(New York: Gordon and Breach).[4] Erdélyi, A., et al., 1953–1955, Higher Transcendental Functions, Vol. 2 (New York: McGraw-Hill).[5] Marichev, O.I., 1982, Handbook of Integrals Transforms of Higher Transcendental Functions (Chichester: Ellis
Horwood).[6] Prudnikov, A.P., Brychkov, Yu.A. and Marichev, O.I., 1990, More Special Functions. Vol. 3. Integrals and Series,
