Definite Integrals for Expressing

Polynomials into Fourier Series

Bennosuke TANIMOTO*, Dr. Eng. & Dr. Sci.

(Received October 10, 196e)

Synopsis. This article gives several !<inds of definite integrals, which will be con-

venient to express polynornials into Fourier series. Examples are added for illustration.

We take the following polynomial of p:

w(p) =ae -= aip+a2p2+-･+arpr, (1)

which isdefined in the interval (O,1>; ao, ai,･･･, ar being constants independent

of p. This polynorrLial may be writteR in the form

cx) w(p) =Xcn sin nzp, (2) n--1

or in the form

co w(p) =co +Zcn cos nzp. (3) n==1

The present tables will provide a convenient means for expressing equation

(1) into equation (2) or (3).

For instance, we take simple beams such as given in Figs.ItN,4. Figs.1

and 2 are subjected to "continuous" loads throughout the whole interval (1,O),

and hence Tables ItN-II serve for such cases. Fig.3 is subjected to two kinds

of loads, which is connected at the midpoint of the beam; and hence Tables

III--VI serve for such a case. Fig.4 is subjected to a partial load in a certain

range of the beam; and hence Tables VIIA"XII serve for s･uch a case. Several

examples which follow will serve as illustration.

Example1 <Fig.l). By the elementary theory of banding of beam, the

defiection w at poiRt x of the simple beam subjected to uniform Ioad q is given

by the equation

:S Professor of Civil Engineerlng, Faculty of Engineering, Shinslva University, Nagano,

Japan.

No. 10 Definite Integrals for Expressing

q

x･tv

Polynomials into Fourier Series

'

q

tX+zva

Fig. 1.

aa 22

o

xtv a

Fig.' 3.

.-,gZ`,(:- 21

where El is the flexural rigldity, and is

the length of the beam. To get the

put

tlli.ilicnsinnarrX..24qEa`i(:

multiply by sin nanv dx,on both sides,

Putting

fi = p, du ==

a

we have

(Integration of left-haRd side) =: S," (.O,O.nt-,

qa` <Integration of right-hand side) == 24EI

qa4

24EI

and hence

qa4 .1 Cn =2 24EI

Fig. 2.

B

67

･-･- cr-

X4 2- "5 - a4

supposed

Fourier

and intecrrate

a dp,

:E] c2n sin ManX) sin fZaffX -- u:'- cn,

S,a (: - 2tts3 + tl:t`) sin '7.ZX dx

i aS (p - 2p3 + p`) sin nr,pdp, o

5o (s) - 2p3 n- R`) sin nffp dp.

x--" lv a Fig. 4.

to be a constant throughout

sine series for equation (4), we

,.es3 -i- 2111)･

. the result fromOto a,

(5)

In virtue of Table I, we have

1 - (n = 1, 3, 5, ･･･), 7Zr, S: p sin nrrp dp ...

I - - (n == 2, 4, 6, ･･･); nrr

l6 i ffT- <nx)3 <n "= 1, 3, 5, ･･･), So p3 sin nzp dp ..r

}6 - 5itirr + <nz>3 (n = 2, 4, 6, ･･･);

112 48 i n". - (nrr)3+ (n.>s (n=1, 3, 5, ･･`), Sop` sin nap dp ,.

112 rm' iiTT + (nrr>3 (n = 2, 4, 6, ･･･).

Substituting these values into equation (5), we get

11 48 ,-,.(1 -2+ 1)+ (..)3[(th 2) × (- 6) - 12} + <nn)s qaa

× (n -- 1, 3, 5, ･->,cft = 12E7 11 iit}T(- l+2- 1) -i- <n.)3[(m 2) ×6+ 12} (n = 2, 4, 6, ･･･>

qa` 48 4qa` 1 == 12EJ' (nn)5 =: nsEI'fiT, <7Z = 1, 3, 5, ･'･).

Hence the wanted series becomes

w == 2a4aiiii (: - 2IIiil + illf) =- 4.q,{ii .=i$, II,,,, ... .-i, sin n.rrX. (6)

It wili be seen that the Fourier coeflicient cn is in general given by the equa-

tion

i cn == 2S, w(p) sin nrrp dp. (7)

Example2 (Fig.2). The deflection w of the beam of Fig.2is given by

Ne.10 Deimite lntegrals for Expressing Polynomials into Fourier Series 69

w=36qo"S-I-(7Z-lolilP+3}?). <s)

Referring to Table I, we at once have

w"= ;gifli`i ;,l].-l.iMni,)""isin naTX. (g)

Computation work negessary for getting equation <9) is rnerely

(- 1)n+i;.(7 - 10 + 3) =: O, <- 1)n+i(n}),{(- 10) × (- 6) +3 × (- 20)} = O,

(- i)n+'(ni.), ×3× i2o ×2× 36qoaiiii == 3,qEa ･(- in),""i, == c.;

and this work is effected ethciently by using small pieces of paper of about 5 mm

× 15 mm size, on respective sheets of which the numerals 7, - 10, 3 are written.

Example3 (Fig.3). The defiection w for the beam indicated in Fig.3is

given by the equations

xx3 wi in- g6qo"iiil(25a-4oEr,+!62?) (o<x<ba), (lo)

w2 = g6qo"b(i " is: + 4olil, - i2ot;t3 + soir - i61ilP) '(ga <x< a). (m

In this case equation (7) becomes

' cn = 2S: w(p) sin nmp dp = 2Si"' zvi(p) sin nrrp dR + 2Sl w2(p) sin nzp dp.

For two integrals ef the right-hand side, reference may be made to Tables

III and V respective!y. In this way we get

ttl 8qa" (- 1) 2 Cn = T6EI' n6 <n == 1, 3, 5, ･..).

Thus we arrive at the wanted series

t(ww-rm1 w ;=: £,6qEai;.tll;l3,s.... (-ni6) 2 sin 7i:X. (i2)

ee

70 B. TANIMoTo Example 4. The present integral tables may be used for converting

Fourier .qeries into its poiynomial. As asimple example, we take the

slne serles

W(P) == .x#, 3, s,... (fe.1.,)s Sin n"P (O<P< 1),

and shall find its polynomial. Now from Table I, we can put

ZV (,o) =: cro + aip H- cv2R2 + cr3p3 + a4p4,

where ao, cri, ･･･, cr4 are constants to be determined. Referring to Table

1collecting same powers of we have <nz)r'

2cro -Y cr1 + cr2 + cr3 + cr4 == O, 4a2 - 6crs - 12a4 == O, l <n = 1, 3, 5, ･･･);

48a4 := i, t

6rmcr,crX12crcr2, 1-cro3, ww cra rO' l (n .. 2, 4, 6, ...);

from which

l 11 crO=O, cr1 nd- -g6, cr2=O, a3 =:-li{g, cra={}Tt･

Hence the wanted polynomial becomes

I W<P) == {}itr (p - 2p3 -F p`) (o < p< 1> .

This is the reverse of Example 1.

a

F

No. 10

glven

ourler

(13)

I, and

<14)

No. 10 Defuite Integrals for Expressing

ART. 1 L So pr sin nrcp ap

II. I:prcosnflpdR

IIL S,Sprsinnffpd,o

IV･ Io"'prcosnnpdp

V. Sib pr sin nnp dy

i VL Ss pr cos nmp dp

vll. !i pr sin nrrp dp

VIIL S,e pr cos nup dp

IX SI pr sin nrrp dp

' X' Se? Pr cos nTp dp

XL Si pr sin nffp dp

XII. Si pr cos nTp dp

Polynomials into

Contents

(r == O, i, 2, 3, ･･･;

Fourier Series

PAGE

71

72

73

75

77

79

81

82

83

84

85

87

n = l, 2, 3, ･･･)

71

72

i･ S

B. TANIMOTO

1

pr sin nrrp dp

o. -

No. 10

III

l/

I

Il

r

o

/tt

IiI..j

n

1, 3, 5,･-

2, 4, 6, ･･･

t ttt S] Rr sin nzp dp

ttttttt-ttttt"ttttt tt ... .t.

L

l..I

iiir.

r

E

n

o

iiji

i

-I

l

i

1, 3, 5, -･

tt2, 4, 6, ･･･

s1

pr sin nTp dpo

1

IlI

I

1

itII

2

1

nrc

1 m

l

E

1-- (- 1)n+1pm nrc

1, 3, 5, ･･･

2, 4, 6, ･･･

3

4

5

l

IrlII-

1

li

6

o,2,4,

2s

l 1, 3, 5, ･･･

2, 4, 6, ･･･

1, 3, 5, ･･･

2, 4, 6, ･･･

l, 3, 5, ･･･

2, 4, 6, ･･･

1, 3, 5, ･･･

2, 4, 6, ･･･

--------t---

1, 3, 5, ･･･

2, 4, 6, ･･･

1

nrr

4(nx)3

l

nz

1

nrr

6(nrc)3

1

-e + fzr,

6(nT)3

(- IYI+1[1

nn

(n6rr)3 ]

1

nrc

12 +(nrt)3

48

<?Zff)5

1-- + nrc

12

(nz)3

1

'nrr

20 +(nr, )3

l20

(nx)5

1

-- + 7Zrr .

20

(TZrc)3

120

(nT)5

: (- 1)n+1[ 120nT (nrr),3

+(i.2.0)s]

1nR

.

30

- -+ (M)3

360

(nrr)5

1 440(un

nx)7

1

-- + flZ

30

(nx)3

360

(lzx)5

----------------------------4-

(2s)![ (2s i. nrr - 1

+(2s - 2)! (nz)3

1(2s - 4)! (nrc)5

(- 1)s-i

-F -Y 2! (nre)2s-1

2(- 1)sO!..(er,)2s+i

]

(2s)!rt 1

- +- <2s)! nr,

1(2s - 2)! <nr,)3

1 <2s - 4>! (nff>5

(- 1)s-i

4! (nz)2s-3

+ ...

(nt 1)s

2! (nT)2s-i]

..l

No. 10 Definite Integrals for Expressing Polvnomials ',nto Fourier Series 73

r

1,

3,

5,

2s +l

n

1, 3, 5, ･･･

2, 4, 6, ･･･

si

pr sin nzp dp

(2s + 1)!(

(2s +

1

1>! nrc

(2s + 1)!(um (2s+

11ww le(2s - l)! (n}l5T' + '(2s - 3)! (nx)g

- ･･･ -F 3:, th;)),S,--i, + 1!((i,S,),i-.,]

ttt1

1)! nrr 11+ (2s - 1)! (nrt)3 - Zisrmrrm3')! (f･zrc)5

<- 1)S+i (- l)s + +･-+ i! (nz)2s+1 3! (nre)2s-1

]

II. s1o

pr cos nxp dp

r

o

2

4

5

6

n

1, 3, 5, ･･･

2, 4, 6, ･･･

1, 3, 5, ･･･

2, 4, 6, ･･･

1, 3, 5, ･･･

2, 4, 6, ･･･

1, 3, 5, ･･･

2, 4, 6, ･･･

1, 3, 5, ･･･

2, 4, 6, ･･･

ttt-------t----

s1 pr cos nnp dpe

o

o

2 (7Zrr)2

2(flrc)2 l

.

= (- 1)n 2(nrc>2

r

1

3

n

1, 3, 5, ･･･

2, 4, 6, ･･･

1, 3, 5, ･･･

2, 4, 6, ･･･

s1o

pr cos nrrp dp

2(nz)2

o

3 (nrr)2

3(12rr)2

12+ (71 r, )4

4 (OIT)2

4(nrr)2

24--F (nx)4

24

(nn)4

=: (- 1)n[ 4(7Zrr>2 ww

(:.4),]

5(nrr)2

60+ (nrc)4

240

(nrc)6

5(7ZZ)

60

2 (nn)4

l I

.-m. I i

6 (nrr)2

6(7zT)2

120+ (nrc)4120 +(nrr)4

720 (nx)6

720

(nre)6

E

": (- 1)n[ 6(7z r, )2

120 <fZr,)4

720

(7ZT)61'

i-----i--------i--------------------

74 B. TANIMOTO No. 10

i'

IIl,

Ill

r

j

o,2,4,

2s

1,

3,

5,

2s+1

71

1, 3, 5, ･･･

2, 4, 6, ･･･

1, 3, 5, ･･･

2, 4, 6, ･･･

sg

pr cos nrrp dp

<2s)! [-(2s :'

1

l) ! (nrc)2 11 (2s - 3)! (nr,)` (2s - 5>! (7zT)6

(- 1)s-i <- 1)s +･-+ + 1! (nrc) 3! (nn)2s-2

2s]

(2s)! [(2s - 1

i) ! (nrc)2

1(2s - 3)! (nrc)4 +

+

1<2s - 5>! <nz)6

(ww 1)s +3! <nx) 2s-2

(- 1)s+i

1! <nsc>2s]

(2s +1)! (- 1

(2s) ! (nrt)2

1+ (2s - 2)! <nrr)`

+･-+

1 (2s - 4)! (nn)6

2(- 1)s+i (- 1)s + O! (nr,)2s+22! (nr, )2s

]

(2s +1)! ( (2,) !1

(?Zz)z

1(2s - 2) ! (nrr)4

-ww ---

--F

+

1(2s - 4) ! (ni)6

(- 1)s+i (- 1)s + 2! (nrr)2s4! (nrr)2s-2

]

III. sg

pr sin nTp dp

r

o

1

2

n

1, 5, 9, ･･･

'2, 6, 10, ･･-

sJ,e

R" sin nrrp dp

E 1

nff

2

nrr

1, 5, 9, ･･･

tt2, 6, 10, ･･･

l

1.(nrr)2

I

2nT

1

2

'

'

5, 9, ･･･

lt tt

6, 10, ･･･

1(nrc)

th4nr,

nt 22 <nr,)3

4 (07r,)3

T

o

1

2

n

3, 7, ll, ･･･

'4, 8, 12, ･･･

3, 7, 11, ･･･

tt4, 8, 12, ･･･

3, 7, 11, ･･･

4, 8, 12, ･･･

s.1..

x

o

pr sin nffp dp

1

71r,

o

1<m)r

1

2zZrt

1<s･er,)2

1

4nrt

2<nr, )3 5

]]l

El

No. Ie Definite Integrals for Expressing Polynomials into Fourier Series 75

r

3

4

5

o,

2,

4,

2s

7･z

s/1'

p" sin n-p dp

1, 5, 9, ･･-

i

2, 6, 10, ･･･

1, 5, 9, ･-･

2, 6, IO, ･･+

1, 5, 9, ･･･

2, 6, 10, ･･･

34(nr,)2

6(71r,)4

um.irm

8nrc

12( nr, )2

3

(nr,)3

12-.umnvt..m

ml-(nrc)4

24

(VZrr)5

.116nT

3(71rr)3

48+ (nT>5

516(nrr>

15L' (nrt)4

120+ (71rr)6

l32nrr

5

2(nn)3

60+ (nrc)5

-------b!-------t------t------

r

3

4

5

---

n

3, 7, 11, ･･･

4, 8, 12, ･･･

3, 7, !1, ...

4, 8, 12, ･･･

3, 7, 11, ･･･

4, 8, 12, ･･･

------------

j// pr sin nnp .dp

E

1, 5, 9, ･･･

3, 7, ll, ･･･

]

2, 6, 10, ･･･

1,

.3,

5,

---

2s -- 1

4, 8, 12, -･･

l, 5, 9, ･･･

34<nrc)2

6+ (Tlx>4

1

-- + 8nn 3(flrr)3

1 2(nz)2

'

12(71rr)4

24+ (nrt)5

1 16fzT

3(7Zn)3

5- 16(nrt)2 +

15

(nrc)4

120

(7zx)6

132nn

5 2(nT>3

60

(nrr)5

--------t----------"----------

(2S) ! [ 22s-i (2s

+

- 1)! (nT)2 22s-3(2s - 3)! (nz)`

22s-s(2s ve1 s)! (nn)6 + ' J,' + 2(. -1! l7)iSrrM)ls + ,,(,iF,.lii.,]

(2S)!(ww 22s-i(2s 1 ÷ - 1)! (nn)2'

122s-5(2s - 5)! (nfl)6

122sm3(2s - 3)! (nz>`

(nv 1)s+･"+ + 2.1! (na)2s

o!((irrlil+i]

t

'i

(2s)! [ 122s <2s>! nrr

122sm2(2s - 2>! (nre>3

! -i- "' + ÷ 22s-4(2s - 4) ! (nrc)5

oZ`(7,.lli+'-i]

<2s)! [- 122s (2s)! nrc

+ 122s-2<2s - 2)! (nx)3

1 ÷"-+22s-4(2s - 4) ! (nT)5

22 ' 2 [/ inl.))S2,- "

(2s +1)![ 1

22s <2s)! (nrr)2

+

122s-2(2s - 2) ! (nz)`

1 +."+22s-4(2s - 4)! (nz)6

o!` (irt38--2]

s

I

II

j!t

'

1

Ii[lii

76 B. TANIMOTO No. 10

r

1,

3,

5,

2s+1

n

3, 7, 11, ･･･

2, 6, 10, ･･･

4, 8, 12, ･･･

.1.sg

pr sin nx,o dp

(2s + 1)! (- 122s <2s)! (nz)

i '2 ' 22s-2(2s - 2)! (nz)`

1 -F ･･･ +22s-4(2s - 4)! (nrr)6

(- l)s+i

O! (nT)2s+2]

(2s +1)! [ 1

22s+i(2s +

+

l1)! nrr 22s-i (2s - 1)! (nrc)3

1 ÷-･+ 22sut3(2s - 3)! (nx)5

(- 1>s

2･1! (nT)2s+1)J

(2s + 1)! [- 122s+1(2s +

1 +･1)! nrr 22s-i(2s - l>! (n-)3

1 (- 1)s+i +'"+22snt3(2s - 3>! (fzr,>5 2･1! (nx)2s+i]

IV. sgo

pr cos nnR dp

r

o

l

2

3

n

1, 5, 9, ･･･

2, 6, 10, ･･･

1, 5, 9, ･-･

2, 6, 10, ･･･

1, 5, 9, ･･.

2, 6, 10, ･･･

1, 5, 9, ･･･

2, 6, 10, ･･･

s"o

pr cos nrrp dp

1

nx

o

l2nr,

1(nrr)2

2

<nR)2

1

4nn

2(nrr)3

ww 1 (flrr)2

18nre

---{l----

(?Zn)3

6+ m.ttttttumtL t

<7Zz)4

3'

4(nrr)2

12

(nr, )4

r

o

1

2

3

n

3, 7, 11, ･･･

4, 8, 12, ･･･

3, 7, 11, ･.t

4, 8, 12, ･･･

3, 7, 11, ･･･

4, 8, 12, ･･･

3, 7, 11, ･･･

4, 8, 12, ･･･

ss'

pr cos nnp dp

l

nrr

o

1

2nz-1 (elff)2

o

1

4nrc

2+ (nrr)3

1

.(IZz)2

1

-- + 8nn

LT3...ve-

(1･trr)3

6'･- (?Zn)4

34(nT)2

No. 10 Definite Integrals for Expressing Polynomials into Fourier Series 77t/,･

fF

t r

4

o,2,4,

2s

1,

3,

5,

2s +1

n

1, 5, 9, ･･･

2, 6, 10, ･･･

i-----------

1, 5, 9, ･･･

3, 7, 11, ･･･

2, 6, 10, ･･･

4, 8, 12, ･:･

1, 5, 9, ･･･

3, 7, 11, ･･･

2, 6, 10, ･･-

s"o

pr cos nrrp dp

1

16nre

3-- + (7Zrr)3

24 -

(nff)5

12(nrr)2

12+ (nn>4

t--------------i-------l

T

4

1･Z

3, 7, ll, ･･･

4, 8, 12, ･･･

------------

s8'

pr cos mp dp

1

-÷ l6nrr 3-(71x>3

24

(nz>5

12(nit)2

12

(nx)4

-i-t-t--------i---------

(2s)![ 122s (2s)! nz

122s-2(2s - 2)! (nn)3

1 + 22sm4(2s - 4)! (nrr)5 + --- +

(rm 1)s

O! (nre)2s+1]

(2s)![um 122s (2s)! 7tz

1-iun 22s->(2s - 2)! (nT)3

1 22s-4(2s - 4)! (nrc)5 +".+

(- 1>s+i

O! (nrt)2s+1]

(2s)![- 11 + 22s-3<2s - 3)! (nrr)`22smi(2s - 1)! (nrc)2

1 - 22S'5(2s - 5)! (nrr)6 + ''' +

(- 1)s

2. 1 ! (nrr)2s]

(2s)! [ 122s-1(2s - 1) ! (nx)2

+

1 22sm3(2s - 3)! (nrr)`

122s-5' (2s - 5)! (nrc)6 +

ttt(2s ÷ 1)! [22,+i(2s e 1>! nr, -

1 + 22sne3(2s - 3)! (nrr)5

." +(- 1)s+i

2･1!(nz)

2s]

122smi (2s - 1)! (nT)3

<- 1)s+ '- rl- un + 2. 1 ! <nrr)2s.+1

(- 1)s+i

O! (nz)2s+2]

'(2s + 1)!C- 2,,+,(2s lli 1)! 7z.

1 22s'3(2s - 3)! (nx)5

+

+

122s"i(2s - O! (nrti'3

(- 1)s+i"' + + 2･1! (nn)2s +1

(- 1>S+i

O! <nr)2s+2]

(2s -F

t tti)! [- nEis<28!nv(nT)2 ÷ 22s:i(2g']l' '2)'! (iiT)`

1 22sww4(2s - 4)! (nr,)G ". +

(- 1)s +22･2!(nz)2s

2<- 1)s+i

O! (nr,)2s+2]

78 B. TANIMOTO No. 10

r

1,

3, 5,

2s+1

n

4, 8, i2, ･･･

s"o

pr cos nmp dp

(2s +i)! C 1 1

22s (2s>! <nn)2

+

22s-2<2s - 2)! (nr,)`

l ., m... +

<- 1)s+i

22sww4<2s - 4)! (nz>6l 22 . 2 ! (nrc)2s

]

v. s1gpr sin nap dp

r

o

1

2

3

4

11

1, 5, 9, ･･･

2, 6, IO, ･･･

1, 5, 9, ･-･

2, 6, 10, ･･･

1, 5, 9, ･･･

2, 6, 10, ･-･

1, 5, 9, ･･･

2, 6, IO, ･･･

1, 5, 9, ･･･

2, 6, 10, ･･･

s1Spr sin nxp dp

1

nn

2

nx

1

nz

1(nn)2

3

2nn

1

nz

l(nre)2

2(nT)3

5

4m1

nrc

4+ (nre)3 ' 34(nn)2

t tttttwwt.tt

6 (flz>3

6-1- - (OZT)4

9

--- + 8nfi

9(nft)3

1

nr,

12(nr, >2

12+ -- (7Zre)4

12 (71.rr)3

24" (7-lrr)5

17- -i- 16nr,

15

(fiZrt)3

48<fln)5

r

o

1

n

3, 7, ll, ･･･

4, 8, 12, ･･･

3, 7, 11, ･･･

4, 8, 12, ･･･

2

3

4

3, 7, 11, ･･･

4, 8, 12, ･･･

3, 7, 11, ･･･

4, 8, 12, ･･･

3, 7, ll, ･+･

4, 8, 12, ･･･

s1 pr sin nxp dp"

i

nn

o

1

-÷m 1(nrt)2

1

2nT

1

-+nrt (nn)2 <nr,)3

3

4nr,

1

nx

3+ 4(nrr)2

e h --m 6 (nz)3

6 (nr, )4

7

8nr,

3÷ (nz)3

1

-+TZrr

12(nx)2

12

(nrr)4

12 (nfl)3

24"- (nx)5

15

--- + 16nrr

9(7trc)3

No, 10 Defuiite Integrals for Expressing P61ynomials into F ourler Series 79

r

5

---

o,2,4,

2s

1,

3,

5,

2s -F 1

71

1, 5, 9, ･･･

2, 6, 10, ･･･

-------t--i-

1, 5, 9, ･･･

3, 7, 11, ･･･

2, 6, 10, ･･･

4, 8, 12, ･･･

1, 5, 9, ･-+

1

fndtT

+

Sl pr sin nrrR dp

- 5 16<nx)2

15 120-- + (nT)5(71re)4

20

(nx)3

120 (nz)6

33 45- 32nz + nv2'

(nz)3

180

(7zz)5

----i-------------------

pt

5

n

3, 7, 11, ･･･

4, 8, 12, ･･･

------------

Sl,,

1

iit}Ir,

pr sin nmp dp

5

-F - 16(nrr)2

15 120 +(7Zff)4 (7Zfl)5

20(?lrc)3

120÷ ------- (11r,)6.

31

32nz

35mi- umrmm" 2(nr, )3

60

(nff)5

------------------------

(2s)![

(2s)! nT 22s-i <2s -

1+ 22sm3(2s - 3)! (nrr)` +

11)! (nz)2 (2s - 2)! (nT)3

(2s - 4)! (nrr)5 22s-5 (2s - 5)! (nn)6

(- 1)s (- 1>s - --- + ---um,.wu..unne + O! (nz>2s+1 2･1! (nrr)2s

]

(2S)!( (2si nz ÷ 22s-i(2s -i

i 22s-3(2s - 3)! (nrc)`

11)! (nz)2 (2s - 2)! (nrr>3

+(2s - 4)! (nrr)5 22s-5(2s - 5)! (nrr)6

(- 1)s+i (um 1)s

-･-+ + O! (nrc)2s+1 2. 1! (nrr)2s]

(2S)! [' 22?2i2s+)!lnrr 22s-2 + 1÷ 22sm2(2s - 2)! (nx)3

22s-4 + 1

22s-4(2s - 4)! (nn)5"- +

2(- 1)s+i

O! (nrr)2s+1]

(2s)! [- 22s - 1 -+ 22s <2s)! nr,

ttttttttttttt

i)![ (2s -i- 1)!7zr, -

i22s-2(2s - 2)! <nn)"

22s-2 - 1

22sum2(2s - 2)! (nx)3

22s-4 " 1

22s-4(2s - 4)! (nr,)5 + tt ttt t t

(- 1)s(22 - 1)

22.2! (nr,)2s-1]

(2s +

---

22s(2s)! (nrr)2 (2s - 1)! (nrr)3

22s-4(2s - 4) ! (nrc)6 (2s - 3)! (nrr)5

(- 1)s+i (- 1)s - ''' + 1! (nrr)2s+1 + o! (nz)2sTt ]

80 B. TANIMOTO No. 10

r ff'

I

1,

3,

5,

2s+1

3, 7, ll, ･･･

2, 6, 10, ･･･

4, 8, 12, ･･･

Sl,.

pr sin nr,p dp

(2s + 1)!((2s + l + 1)! nx

1 22s-2(2s - 2) ! (nrt)`

122s (2s>! (nrr)2

1 <2s - 3) ! (nr, )5

m '" -i- 1!

1 (2s - 1)! (nn)3

1+ 22s-4(2s - 4)! (nz)6

(ww l)s (- 1)s +(nx)2s+1 O! (nz)2s-F2

]

(2s + l)!(- 22s+1+1 22s-1+1 + 22s-i(2s - 1)! (nz)322s+i(2s + 1)! nrr

22s-3 + 1 3(- 1)s+i

- +".+ (nz)5 22s-3<2s - 3)! 2･1! (nff)2s--1

)j

(2s + 1)![- 22s÷1-1 22s-1-1 + 22s-i(2s - 1)! (nT)322s--i(2s + 1)! nn

22s-3 - 1 - 22s-3(2s - 3)! (nz)5 + ''' +

<-2･1!

1)s-

(nn)2s+1)

Vi･ S 1

pr cos nmp dpl

r

o

l

2

3

11 sL

pr cos nTp dp

1, 5, 9, ･･･

2, 6, 10, ･･･

!, 5, 9, ･･･

2, 6, 10, ･･･

1, 5, 9, ･･･

2, 6, IO, ･･･

1, 5, 9, ･･･

1

nn

o

ll2nfl (nn)2

2(nrc)L)

1

4nn

2- ･+ (f'Zz>2

2(nrc)3

3ewt)2

138nrr (nrr)2 +

+

3 (nT)3

6(fZrr)4

r

o

1

2

3

o･f.

3, 7, 11, ･･･

4, 8, 12, ･･･

3, 7, 11, ･･･

4, 8, 12, -･･

3, 7, 11, ･･-

4, 8, 12, t･･

3, 7, 11, ･･･

s1

gpr cos nrcp dp

1

mo

i

2nrr

1(nrr>2

o

1

4nn

2(na)2

2(nrr>3

I(71T)2

l

8nn

3(nz>2

+

3(nz)3

6(nff>4

iE

Ne. Ie Definite Integrals for Expressing Polynomials into Fourier Series 81

Ii

I:l

i

r

3

4

o,

2,

4,

2s

1,

3,

5,

2s +1

n

2, 6, 10, ･･･

1s pr cos nr,p dpth

l

-15 - 12ww4(nrc>2 (nrr)4

1, 5, 9, ･･･

2, 6, 10, ･･･

------------

14l6nn (nx)2

24 + (?Zrc)4 e

3÷ (nT)3

24 (nT)5

9362(nrr)2 <nrt)4

---l---i---------------l

r

3

4

1･Z

4, 8, 12, ･･･

3, 7, ll, ･･･

4, 8, 12, ･･･

--------i---

Sl, pr cos nmp dp

94(nz)2

1, 5, 9, ･･･

3, 7, 11, ･･･

2, 6, 10, ･･･

4, 8, 12, ･-･

1, 5, 9, ･･･

16nn (nrt)2

24 ++ (7Zz)4

3(nT)3

24(nrr)5

7 122<n rr)2 <nT)`

------------------------

(2S) ! [- 22s <is>! flz

1+ (2s - 3)! (nrr)4

1(2s - 1)! <nn)2

122s-4(2s - 4)!

+"

1+ 22s-2(2s - 2)! (nz)3

1(nrc)5 (2s - 5)! (nn)6

<- 1)s (- 1)s+i' + 1! (flz)2S + O! (nn)2s+1 ]

(2s)![ 22s(2is)!nrr

i+ <2s - 3)! (nrr)4 +

l(2s - !)! <nn)2

122s-4(2s - 4)!

l 22sm2(2s - 2)! (nrc)3

1(nrc)5 (2s - 5)! (nT)6

(- 1)s (- 1)s." + + 1! (nT)2s O! (nT>2s+1

)(2s) ! [ 22,-?iS2-s i-+ 1

1)! (nT)2

+

22s-3 + 1

22sm3(2s - 3) ! (nrr)4

22s-5 + 1

22s-5(2s - 5)! (nT)6

3(- 1)s+i"' 't- 2.1! (nrt) 2s

](2s)!(2,,-,2(2 gi-im 1

1)! (nn)2

+

22s-3 . 1

22s-3(2s - 3)! (nz)`

22s-5 nv 1

22s-5 (2s - 5) -."+･ (nz)6

(- 1)s+i

2･1! (nn)2s]

(2s + 1)!C- 22s+i(2is + i)! nz m (2s>!i(n-)2

22s-i (2s - (7ZT>3 <2s - 2)! (nff>4 1)! 22S-3(2s - 3)! (nr,)5

rm..(2s:.Al.Iwgwn.I)s. t..'':2･i(i(,i)l2"Lli mF 6!um(,i.)ee,1 pt

82 B. TANIMOTO No. 10

i: 1'

1,

3,

5,

2s +1

7¢

3, 7, 11, ･･･

2, 6, 10, ･･･

4, 8, 12, ･･･

Il, pr cos nrrp dp

'

(2S + 1)! ( 22s÷i(2sl+ 1)! nr, - (2s)!i(nrr)2

(2s - 2)! (nz)4 22s-3(2s - 3)! (nr,)5 1)! (m)3 22s-i(2s -

- 1. m".+ (- 1)s + (- 1)s+1 (2s - 4)! (nr,)6 2ol! (nT)2s+1 O! (nrr)2s+2

]

(2s +l)! [ 22s 22s + 1

<2s)! (nz)2

+

22s-2 + 1

22s-2(2s - 2)! (nz)`

22s-4 + 1 m".÷22srm4(2s - 4>! (nn>6

2(- 1)s

O! (nrr)2s+2]

(2s +1)! ( 22s - 1

22s (2s)! (nn)2

+

22s-2 m 1

22s-2(2s - 2)! (nn)`

22sm4 ffm- l

22sm4(2s - 4)! (nz)6--- ÷

3(- os+i22.2! (nrr)2s

>j

vll. Si pr sin nxp dp

r

o

1

2

3

4

"

l, 2, 3, ･･･

si

pr sin nzp dp

1

nx(i - cos nrre)

1, 2, 3, ･･･

1, 2, 3, ･･･

1, 2, 3, ･･･

!, 2, 3, ･･･

------------

1

nrrg cos nr,e +

1 sin nz6(nn)2

l

7Zrr

g2 cos nr,g S- 2 S sin nr,6 -(m)

2(11re>3

(1 - cos nr,g")

1. pte3 nr,

cos nrre 3rr

1wu (nfl)2 g"2 sin nag" 6+ -------･ (TZrr)3

a cos nrr6 --6-L

(nrr)4Sln 7zrgr

1- -e4 i¢z

COS nzg" lhft 4(nrc)2

' '12g3 sin nrrg + --- g"2 cos nzg" (TZrr)3

24 . 24 - -- 6sin nzg + <nz)4 (nr, )5

(1 - cos nne)

---t-t---tt---------i---

l

I

No. 10 Definite Integrals for Expressjng Polvnomials into Fourier Series 83

r

o,2,4,

2s

i'

3,

5,

2s+1

n

1, 2, 3, ･･･

1, 2, 3, ･･･

Si pr sin nTp dp

<2s) ! [- e2s-1 sin nrrge2s cos ozrrg e2s-2 cos nz8 -+ + (2S)! 11T (2s - 1)! (nrc>2 (2s - 2) ! (nrr)3

e2s-3 sin nxg e2s-4 cos ntf e2sm5 sin nrre

± <2s- 3)! (nn)` (2s- 4)! (nrc)5 (2s - 5)! (nx)6

+ ... rrt- (- 1)swwie.e.in nftg"' ntiun (- 1)s(1 - cos nT6)

O! (nx) 2s +1 1! (nr,)2s]

(2s + 1)! [- e2s+i cos nr,g" 82S sin nTslt e2s-1 cos nrrg ----- -y + (2s)!(nT)2 <2s + 1)! i･zn (2s - D! (nz)3

62s-2 sin n-$-" 62s-3 cos nffse 62s-4 sin nr,e- -tvet-ummmttttttttttttt.ttt.t. ..L..t m ..u-ttttttttrmtt..-utt.tttttttumtt + ttttt tttwuvettttttrm"

(2s - 2)! (nrc,)4 (2s - 3)! (nn)5 (2s - 4)! (nT)6

tttt -F...+(wwi),Sti,20,2,nTe+L-,,,i,).'Xli.I},il.tttS]

VIII.Si pr cos nmp dp

:

r

o

n

1, 2, 3, ･･･

Si pr cos nzp dp

1.- sln nrrenff

1

2

3

4

1, 2, 3, ･･･

1, 2, 3, -･･

1, 2, 3, ･･･

1, 2, 3, ･`･

'" l l I

1

nrr6 sin nTe -

lr(n.)2 (1 H COs nffg")

1

rmg2

f･zff

sin nrrg- + sin nne 6 cos me -(IZn)2 (fZr,)3

1 A3---gnx

sin nffg-+ A--.e2cosnzg- -6-ssin me a- 6 ( 1 - cos fzTe) (nrr)2 (nrr>3 (nr, )4

1 fi4-gnr,

sin nze + 12 4 e2 sin nz8 g-3 cos nne - (nz)3(fZx>2

24 24 - (n.>4 g COS nrr6 + (n.), sin nrcc".

I------lp----------t-----

84 B. TANIMOTO No. 10

I

rl i

n

11LI

sj

pr cos nrcp dp

o,2,4,

IZs

r

f[

E

1,

3,

5,

2s +1

1, 2, 3, ･･･

1, 2, 3, ･･･

(2s)! [ e2s sin nr,g e2s-i cos nr,e e2s-2 sin nrrg"

+- (nz)2 (2s - 1)! (2s)!nr, (2s - 2) ! (ozrc)3

e2Sww3 cos nffg- g"2sm4 sin nffe e2s-5 cos nzg (Vzrr･,)4 (2s - 3>! (2s - 5)! (nT)6 <nrc)5 (2s - 4)!

(- 1)s sin nrr6 (- 1)snti gfi cos nzg -".+ + O! (nrc)2s+1 1! (nrr)2s

hj

(2s ÷1)!

cg,2i{, sX,",1".,6 + i2g,sgi,:'#'i-t,{ - ,gfiIS--iE?ge, n,.-.6,,

62s-3 sin nTsr"- e2s-4 cos nTg" e2s-2 cos nx.h"

(nz)s <nT>4 (2s - 4)! <nx>6 (2s - 2)! (2s - 3)!

(- 1)sg sin nzg" ( -- 1) s+i(1 - cos nffg)

-". -F + 1! (nr,)2s+1 O! (nrc>2s+2]

IX. svg

prsin nTp dp

r

o

1

2

3

4

17

1, 2, 3, -･･

1, 2, 3, ･･･

1, 2, 3, ･･･

1, 2, 3, ･･･

1, 2, 3, ･･･

sr.

pr sin nTp dp

1

7Zre

(cos nzg - cos nzrp)

1

nrc

(g cos nrr.i - rp cos nTv) -1 (nx>2

(sin nrcg" - sin nr, rp)

1. (g27ZT

cos nzg - q2 cos nap> 2(fZz)2

(g sin nrrg - q sin nTq)

2 - (n.)3 (COS nntsft - cos nzny)

1

nT(g3 cO.

3 (g2 sin nrre - v2 sin nrrrp)nrce - ny3 cos nrcrp) - <nz)2

<sin nxg - sin nrcrp) <g cos nscg - rp cos nrcrp) -V (nrr)4 (nrr)3

1

nrr .4(64 cos 7trrtt - rp4 cos nxrp) - <nrc)2 12 (62 cos fzze - v2 cos nzv) +'7Zr, >3

(g3 sin nrrg"' - rp3 sin nnrp>

24 (e sin nfl.lt - rp sin nnv)(7ZT)4

24 t- (n.>, (COS nrce - cos nTty)

:t

'

---.i...----.

f

i

l

Il

No. Ie Definite Integrals for Expressing Polynomi..Is into Fourier Series 85

r

o,2,4,-s-

2s

n

1, 2, 3, ･･･

!t

3,

5,

2s+1

1, 2, 3, ･･･

si

pr sin nnp dp

(2s)! [

+

- ---

e2s cos nrcg - v2s cos nffrp 62s-l

<2s)! nrc

st2s-2 cos fz.rcE rLrm!z12.i::2LCOS 7Zt.tq +

(2s - 2)! (nT)3

g"2s-4 cos nxe - rp2s-4 cosnzrp

sin ulg - rp2s-i sin nr, rp

+

<2s - l)! <nre)2

g2s-3sin nrre - n2s-3 sin fzrte

(2s - 3)! (nrr)4

g2s-5 sin nng" - )7L'S-5 sin p'zT)7

(2s - 4)! (varc)s

(- 1)s(e sin nffs" - rp sin nrtv)

+

(2s- 5)!(nz)6

(- !)s (cos nxe - cos nTv)

1! (nrc)2s O! (nn)2s+11

(2s + 1>! [ 82s+i cos nrcg - rp2s+i cos nzrp e2s sinnrrg" - rp2s sin nno

<2s + 1)! nrr

e2s-i cos nze - rp2smi cos nxrp

(2s - 1)! (nz)3

e2s-3 cos nreg" - lf2S-3cos nnn

+

(2s)!<nx)2

g2s-2 sin nTe - tr2s-2sin nrrry

+

-･- +

(2s - 3)! (nre>5

(- 1)s (4 cos nzg- - v cos nxrp)

+

(2s - 2>! (nr,)`

g"2s-4 sin nz.lt - rp2Sm4 sin nnv

(2s - 4)! <nfl)6

(- 1)s+i(sin nrcg - sin nzs)

1! (nrc)2s+1 O! (nn)2s+2 ]

x. S:pr cos nTp dp

r

o

i

2

3

n

1, 2, 3, ･･･

1, 2, 3, ･･･

1, 2, 3, ･･･

1, 2, 3, ･･･

sne

pr cos nrcp dp

1

nz<sin nrca - sin nxn)

1

nn(6 sin nz8 - v sin nrcn> -1

<nz)2 (cos nrrG - cos nTrp)

1- -(42 nn

sin naf - rp2 sin nT?) -2 (nz)

, (g cos nTe - ny cos nap)

2 (sin nxe - sin nTv) + (nrr)3

+

!.-(63 sin nzg - u3 siB nnrp) -nn

6. inr,)3 (e Sln fZZts-" - ty sin nrrp>

3 (62(fZff)2

6+ (IZT)4

cos nne - ?2 cos nzv)

(cos nrc r/ - cos nrcrp)

86 B. TANIMOTO No. 10

I.1

r

Il

I.

7Z

4i'

ll･

[ii

I

l -･

l, 2, 3, ･･･

sl

pr cos nrcp dR

o,2,4,

2s

1,

3,

5,

2s+1

1Mn-t(g4

+ 12- <nr, >3

sin nr,g" - if sinnrcry) -

(g"2 sin nr,g" - rp2 sinnn7)

4 , (g3 cos nrr6 - ty3 cos nscv)(nT)

24+ (n.,)4(e COS IZree - rp cosnrcny)

24 ma (n.)s(Sin na4 - sin nTrp)

l-t ny ----l--t---------i---

1, 2, 3, ･･･

1, 2, 3, ･･･

(2s)! [- e2s sin nr.g - ny2S sm ngrp g2s-i cos nzg' - v2s-i cos nop

+

+ ... +

62s-2 sin nr, t'-'

<2s)! nrr

m ry2s-2 sin nzu

(2s -- 2)! (n r, )3

e2s-4 sin nr, gi - ry2S-4 sin nxv

+

(2s - 1)l (nT)2

g2s-3 cos nr, g' - rp2s-3 cos nzrp

(2s - 3)! (nrc)4

g2s-5 cos nxg - o2snt5 cosnxrp

(2s -- 4)! (nT)5

<- 1)s <g" cos nne - v coS nrcv)

+

(2s - 5)! <m)6

<- 1)s+i (sin nffg - sin nff1)

1! (nrr)2s O! (nz)2s+1]

(2s + 1)! [- gfi2s+i sin nr,ein

rp2s+1 sin nffrp e2s cos nffg" - rp2S cos nrcv

+

-l- ."

e2s-i sin nrtg"

(2s + 1)! nrc

ww rp2s-1sinnzty

+

(2s)! (nr, )2

e2s-2 cos nr,g - v2s-2 cos nrrrp

(2s -

e2s-3 sin n"gn'

1)! <nr,)3

rm .ry2s-3 sin nnq

<2s - 2)! (nrr)`

62s-4 cos nr,tf - rp2s-4 cos nz?

+

(2s - 3)! (nz)5

(- 1)s+i(5sinnne - 9 Sln 11T, rp>

+

(2s - 4)! (nx)6

(- 1)s+i<cosnrrgrv' -

1 ! (nrc)2s+1 O! (nre)2s+2

cos nffrp>]i

Xi･ S1

v

pr sin nnp dp

r

e

1

n

1, 3, 5, ･･･

2, 4, 6, -･･

s1

v

pr sin nr, ,D dp

1

-(cos nrrn ÷ 1)nx

1

-(cos nRij - 1)?1rc

I

l, 3, 5, ･･･1

-･(lf cos nr, ij

7'Zr,

1.{- 1) - Slll <nrr･,)2f7ff K･'

No, 10 Definite Integrals for Expressing Polynomials into Fourier Series 87

r

1

2

3

4

---

o,2,4,---

2s

n

2, 4, 6, ･:･

1, 3, 5, ･･･

2, 4, 6, ･･･

1, 3, 5, ･･･

2, 4, 6, ･･･

1, 3, 5, ･･･

2, 4, 6, ･･･

1, 3, 5, ･･･

2, 4, 6, ･･･

'

Sl pr sin nrcp dp

1

nrc

(rp cos nTty - 1) -1 (Olrr)2

sm nffo

Hl,-

nrr

(v2 cos nrr? 2+ }> ww (n.)2V Sin n-u -

2

(nrc),(cos nTq + O

1

nrr

(rp2 cos nrcrp me 2.1) - (n.),q sm nzq2

(ilT>,(Cosnrcny - 1)

1

nrc

(o3 cos nn? +

i'

321) - (nz)2ny sln nzn6

(nrc)3(i7 COS TZrr;7 + 1)

6 + (f7fl)4

sln nrcrp

1

nn<q3 cos nzty - 1)- 32(fZrc)2? sln nmp

6

<nx)3(v cos nr,rp - 1)

6 + (nz)g

sln nrcrp

1

nz(?4 cos nnv +

1) - (.4.)2rp3

12sin nrrq - (nT),(o2 cos nzq + 1)

24 24+ (n.,)4rp Sln nrcrp + (n.,)s(COS nrrrp + 1)

1

nrc

(rp4cos nrcrp - 1) -4

(nrr)2rp 123 sin nr,n - (n.),<v2 cos nnrp - 1)

24 . 24+ (n.)4V SM 7Znrp + (n.),(Cos nnq - 1)

---t-----------------4--

(2s)! [v2s cos nrrrp + 1 (2s>! nn

rp2s-3 sin nrcv

+ <2s - 3)! (n->`

-- -y

rp2s-i sin nrrrp ry2s'2 cos nrcrp + 1

(2s - 1)! (nn>2 <2s - 2)! (nrr)3

+ q2s-4cosnnty + l - rp2Sm5 sinnrcrp

(2s - 5)! (nz)6 (2s - 4) ! <nz)5

(- li!s(rpns.i)n,,nnv+ (- 1)8[C?7:.?,re,O+,+ 1)]

(2s)! [rp2s cos nflv -1 rp2smi sin nrcrp rp2s-2 cos }･tzq -1 (2s)! nz (2s - 1)! <nrr)2 (2s - 2)! (nr, >3 ,

+ (2rp,2Srm-3 s3iln! :i.T.ry), + viSi,`so2):z(q..->,i - <2v,2S-rm5sii!n(L,;,L?,

- ... + (- ii)l (n･./i):,nffrp ÷ (- igS!((c.o.s);z,rc.n, -- i))

i/

88 B. TANIMOTO No. 10

r

1,

3,

5,

2s+1

n

1, 3, 5, ･･･

2, 4, 6, ･･･

s1 pr sin m,p d,ov

(2s + 1)!

+

rp2s+l cos nr,o +c (2s {- 1)l nn

rp2s-2 sin nrrrp

(2s - 2)! (nrr)4

(- +

1 rp2s sin nr,7

(2s)! <nrr)2

+ rp2s-3 cos nz7 +.

<nrr)o (2s - 3)!

1)s(rp cos nrrv + 1)

1 ! (nr,)2s-1

q2s-i cos nr,rp + 1

<2s - 1>! (nz>3

1 o2S-4 sin nx? (2s - 4)! (nrc)6

(- 1)s+l sinnT7+ O! (nz)2s+2

)

i'

i

(2s ÷

1>! [v2s-}-(12csoi? zre)l i.,., 1

rp2s-2 sin nrrty

+ (2s - 2)! (nz)"

(- +

v2s sin nrrrp

(2s)! (nrr)2

rp2s-3 cos nffrp -

+ (2s - 3)! (nrt)51)s (o cos nzrp - 1)

!! (nn>2s+1

n2swwi cos nmp - 1

(2s - 1)! (nrr)3

1 T2s'4 sin nzrp

(2s - 4>! (nz)6

÷ (- &IS,;,ks,ie.#r･rp]

EI

XII. s1v

pr cos nnp dp

r

o

1

2

3

n

1, 3, 5, ･･･

2, 4, 6, ･･･

1, 3, 5, ･･･

2, 4, 6, ･･ ny

1, 3, 5, ･･･

2, 4, 6, ･･･

1, 3, 5, ･･･

N

s1 pr cos nzp dpn

1.- - sm nzrp nz

1.- - sln nzny nrr

1.- -v sln nrco - nT

l(nff>

,(COs nnlj + 1)

1.' ----n sln nrc? - nT

1(nT)

,(Cos nnrp - 1)

12-npm.q sln nzrp

2- (n.)2(rp COS nffrp + 1) + 2.

(nz)3Sln nfl7

1 2ww - v nT

sln nffrp 2(nx)

2<ty COS fZTrp - 2.1> + (n.,)3Slnnfllf

1.' iit}lnrp3 Sin nffv - 3(nrc

>2(vL' cos nrerp + 1) 6.+ (nrt)3rp Sln fZTo

6 -iM <n.,>4(COS nTty -}- 1>

No. 10 Definite Integrals for Expressing Polynomials into Fourier Series 89

r

3

4

--t

o,2,4,

2s

1,

3,

5,

2s+1

fl

2, 4, 6, ...

s)

pr cos nrrp dp

1ww pmV3 nr,

sln nrrq 3Ozrc)

2(rp2 cos nTv -

1, 3, 5, ･･･

i 1

nrc

rp4 sin nzn4

2(v3 cos nzo +(nrr)

24 4(rp Cos nrrrp +(nrr)

6.1) ÷ v sm nrcn <nz)3

6 (cos nrc? - 1) + (nrr)4

tt

2, 4, 6, ･･･

------------

1, 3, 5, ･･}

+

l) --

1> -

12 o. o- sm nzv(nrr>3

24 . sln nxn(nrc)5

1.- -o4 sin mrp nrc

4

(m)

12,(v3 cos nrcrp - 1) -- (n.),q2

24 + (n.)4(q COS nrco - O

sln nrrrp

24 ve <nz)sSinnrcrp

--!-----t------------i--

(2s)! [- o2s sin nrrrp rp2s-i cos nnrp +1

+rp2s-2 sin nzrp

+

(2s)! nff

rp2s-3 cos "r, rp +1

(2s - 1)! (nn)2

rp2s-4 sin nno

(2s - 2)! (nrc)3

o2srm5 cos nmp + 1

(2s - 3)!

+･-+

(nT)4

(-

(2s - 4)! (nz)5

1)s(ty cos nzry + 1)

+

(2s - 5)! (nn)6

(- 1)s+i sin nrrrp

1 ! <nre)2s O! <nx)2s+1]

2, 4, 6, ･･･

1, 3, 5, ･･･

(2s)! [-

+

ny2s sin nTo q2smi cos nTq - 1 +(2s - 1)! (nff>2

1 rp2sm4 sin nrrry

rp2s-2 sin nrcrp

(2s)! nz

lj2s-3 cos nnrp -

(2s - 2)! (nn)3

ry2s-5 cos nzv - 1

(2s - 3)!

+."+

(nrr)` (2s - 4)! (nx)5

(- 1)s (q cos nzrp - 1)--

<2s- 5)! (nn)6

(de 1)s+isin n rtv

1! (nz)2s O! (nn)2s+1]

(2s + 1)![- v2s+i sin nTo rp2s cos nr;ry + 1-y

ty2s-1 sin nnv

+

(2s + 1)! nrr

rp2s-2 cos nr, ty + i

(2s>1 (nre>2

ty2s-3 sin nrcrp

(2s - O! (nrr)3

ty2s-4 cos nffrp + 1

(2s - 2)! (nz)`

(- +."+

(2s - 3)!

1>s+i rp sin nrr4

1 ! (fzz)2s+l

<71z)5

(-+

(2s - 4)! (nrr>6

1)oS i (ln/L?2S,IZI7 -i" 1]

90

ili"l

i' '''"'

Definite Integrals for Expressing Polynomials into Fourier Series No. 10

n

IItIt

s1

pr cos naR dpv

i

I,

3,

5,

2s+1

l

2, 4, 6, ･･･

(2s +

+

i)!(mu nyimiigi,gi?,IZ?,th " .kS ;o,;."z.z,fi+ ,g,2ili ?,i?,;z,;;,?,

g2s-2 cos nr,ry -1 ep2spm3sin 7zr,v ty2s-4 cos nrq-1

<2s - 2)! (nff)" (2s - 3)! (nn)5 (2s-4)! (?zfl>6

+ ･･･ + (thi/)i;i)rp,,/r,n nnv + (- 1>so+.!i(;c.o>s,,7{n,,o- 1> ]l}