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,SW-#jl .";i t

l+-f

grl ..ti-;l:

sf

_J

6,

A

c -^-^'ol 1l 13:-rr.,tf

J'3t"-.d'l

-Lts

,frt:f(x) '

oq1l

Cb

Ei,t#.s-r.l

]evf(x)

ry

(20)

ffrl=fQ) ,

-l<x<o ip=tl

f(x)

\

" <x</ rLrU ;.r

.li

ts;il

6f t

otj- t-.1ir

,-r--# 6f

6lrbf @).t(ii

,,2,-.1 _,2lrt

f

Al

=

_?*

,:,

an

cos

+,

=

f(x)

i

'\. .

<lcl

o,=*

_I'fOl

,orffxdx

_;

n=

or\ryr....

€,-1,-irr,.;

C;a

ltf6)

tt ,;i3

fro=f@)

,

o<x<t

;P=ll

fg=-f(-x) i

-t<l1.

.:,.,,1

_Alf@)

L .

<rcl

drU;)

₁

r'l:

,r-tL-,..'

t-l!

Gf

6lrbf

@)

f@)

=,l,rn

t,

_+x

=

f(x)

i

c<x<l

b^=+

_{[.'flt)r*+xdx}

_;

_n=\ryrfr...

.+td

lttj.rlrr-t

_\f(x)_Ctrq

pu;-u-t!

_{r/_#t,f*,.-,CV-r}

_.?

_Jlb

(

lk

_{- .}

_{,,,, t}

*i

d,:.a

tslir-lltf(x)

gtu

l:r

I

c<r<-f(x)=

₁

_{* lr_vt. t}

Y

(21)

(rt

) JJI J,.ri

g:bf(x)

t{

_rbtt-

'/H

usP

6r

c;'t-

gtl

f(x)

sin

T, *

₌

f

r

_1")

,in

_+,

dx

+

{,,'

o-,,

,*

+

x dx

l

b^=+

!.t

=#:,

_'T

an=$t

tk,h

wa--an=il *,

svt7x

bn=Sr

_{#,o,ff,[.#sinffx}

f

_=-*

_rl'

_,

-ffit*T

lu, bn=

ftt

-J-ro,

W

+

Lrn+

+

I*'T

;l c..,1 _'>1Vf(x)_\_-b f(x)

sinffx

;

,<l1l

t;

+

't--sinfft=#'*T

V ,f *

+u! c9f

ta"i.ilr

": t(t-') ,o,

y

,

o ft1c I cr-jtf(x) q _rDU:.

,f #

c9r

611.

yh:ao

n llt fa,t ' _'

I

x

cos

T, *

+

!,,

(l-x) cos

T,

*

I

f

nrc

fficosTx

llr + o llr + c

(22)

t(t-il nn It r

nn

lt

,*

stn

_T

x

1,,,-

"v

'ot

t

lu,

I

on=#(rcos+

cosnn-\)

a.=k

&rlJtr

f(x)

-+. #

_:,;!

{r,o,

+

cos

wt-

\) cos

T-r.t -e-,,, q t_ _-ro)Jt'

F-"

.5lrti

_-r,

Cb

4,

.;-1r

,Sf

r5 -r:ilsr, .itiil .1631

,il.

Cb

.rt 6f

g,r.,f o-r.rLt \ JE .lr

.+fr"')t

.., .+

F-"

{tii

dr,,_lr, _CbJh.i,

Jy*a

_+f(x)=

x

i-\<r<\

_rp

= Y

2

(- \

J"*'

sin n nx

tt=l n

.;,-J.,..:-,1 -r--li )4lJ.ruii..,tlrr

_ctrJ[i,

_{Jt y jJx}

=

\ .l)

6rJui.r:-,,1

orLi:-,| +;_f

eVf

_{r) =}

rL;

cSt+

t - )b

)l

,hAi

jl

6Jq-.t rS .:..,,1 c_1.7

l,

_{Cb.i-;l JeV}

+,-s!

6r;i:

_-;lrl

1,*lrr

5,"--,r

L!#t-f-;:

rllSlrr .J56.

I,t _{-r.j.gL_l}ai

I *uL;

i-r.o

d/_l 1

,h

SV -l p=5

r*,l.-,

er-p,Jnii

,5_ .1r

'fisT

n

\lt[a,bJ

+L-U otfy'.r;".btgk* Atdr^,ltl =

_f@)

_;

asxsb

Cb

.\

.qp

Ykl

alxt

drll

jl 4.

tlf

As

,h*

q )-5 jl.;,,ll

_e

jl

l-r+ ,gbrLaU

_,r.rtr

g;.rl-t^; q dl,1 Y 7t

(23)

(

tr

)

_"'JJl J'''i

rb

q/

-r5J=a .;tf cStfrl.1) r:^.-,

t

L..jtf

Jtlt

.rlr

.+t

A J o),1 t:at,.'t

bdrU

c"-'.1, _J_\,.7)9J.r, gl-llr

lr.j!

cgtrrl

trl r(-,

cr.llt

r., U- .r5.r, _,J.-

.f b

_L|,J'>

.J-lb

'rrLr

lt'.,-2-t _[a,b]

_af

,6]

Ji.b o.;1..-r

6h* ltl

= f@)

;

asxsb _{Cb .Y}

.-;;

dt{Jl

g

5

-rr-l-fu

4.-.J^t

st ;-Fb,5l,r&U -$

dlilr

bV

9f

t

o)!..J11*

.!iU

r;-1, _J

_\;3

C,,|,..q:.:,

g;bl);

r5Lr&ti

.€e.Jbl

L

J31;J

6rlf*'r1jJ-,itd

jl

#

r!U6l^*

ca,2,.t-rrb/ = g@)

;

asx<boKy'

.(.-Q-

_dt#-J

_d)

\

\A

.63i lim I-o

I0)

I

b

=l

b g(x) sin

k

dx

₌

_" g(x) sin

k

dx

gie.sLJl

(\ ₎

4D=

I sin )'x dx

*+)

G) s-fr

-t"

to

_^s(r*1

T

c'\-- I

,*t-r

it

JlFl

.1rr +

₊

q

r,!r.t

tl (Y)

f;l:

(Y) _-r(\ ₎

_d'trt

_g.+t:

b

u-{

rI(I)

=

["- s@ sin )x

dx

_{

*r

₁ s& sin

k

dx

(24)

eJd

cr: ? b.s-tLi

d-h

JV

rr(1)=-

_I"

g@*

|

lri,,t"*

*1u

ggrrrr**

.

f

'"Is@-s.,+

+

)J siilxdx

"'+ ^

t'f

a

r-rr Lgljl

!

e-.r-y

e.rl;+

;;

_tlt;\Sg(x),Gil

_{€ ,?f}

u;4,r^,lt

lt7@)

{#tt"

+a#

_;olg6)l

s

U

_,',;lr

M

tc,:Vr:,tr b

t!

_u.f

I

{"

₊

s&+

f

)rint"*

I

=

I

["

*

_f

s@)sinJ.xdx I

.oM

), J-J+-a J s(x) siilxdx

t

₌₄

1

till jl

l s@

's(x

+ 4+^:i _J)

l4D

I

Clr5';

u34gl-r Slt6y

#jlJ-b

q;;

.L"tcfcrt,ilq

qi,ai

_{,rlc._f}

_{

d.,al'i .ta- _4;Vl,..oU:l

_4,

_-f

_cgll_b

_*ii

_c.-,1

,Sg

.rh

Al

Sll.r:,-,1

j-tt

| \^)

t

_=ry.

+

_{'

*,

s&) . s@

*

_#

)

| dx

lF-

_{t* q{rti}

_alas@)_cr-t_rtf.:-,1 ,,:-',3l-[a,b] E,."1d.rU

_$(x)

13-l u. _rbt

.

f-Jl

Jli

g st*

lrl. _{,'U,-6)Jre c.-L})Jr

y'

,sl;l q

i+i

r)

.)j

JrlF

.:.r'r4 xrt.s5l

q,ryq

s

,

XlX.-t,sljlq.tr-s

ftt.#

+9=,

Y Y

(25)

(r0)

JJlJ".i

.C=5rlF ,-:,-.,1 tu3-t

Jilg

6 [a,b]

Yrr ./.rb !.lrrl*

?Jfs

5.215-t.5h* [-n ,

n]

ta{Lrt!

tf@'6y'

'Y

4';i

x

_r,

gl;l q

.t*i

r-.lb

!

+

i

(an cos ru

*

bn sin ru) =

_]

fA*)

+ f(x-)

l

Y

n=r .r, Cb

fr

tr:-,,|,

);J-

-tl-t

4t

+f@-) sf@+) s

-[sl

*-lt'bn

tan ..li -;:

6

..f::....OX af-ti.j

€eroGy'.gLJl

s* (x) =- A. ?. Y k

7,ro"

cos nx

*

bn sin

w)

\ ,t

I

"fAl

_cos_n_{t dt}

, *

!"

ral

sin nt dt

d)-Scr-Fu

q.6"i

p*t-.f bn _tan_13t?q

4i

₊

tf

!"fftlcosnt dtlcosruq

Ihrl

sin nt dtl sin rtx}

-Jt

s*(x)=+

i

_fato,**j,

+

_*n'l+

k +

2

cosnt cosra ,8= I s*(x)=

*

sin

nt

sin

wl

dt

(26)

s*(x)=

₊

I"ft,l/ +

*,f,

cosn (t - x)

I

dt

$in

gcos

_{na =}sin (n

+*r,

- sin _tn-

| I,

grbqqitl

t'*t'r

wingr+.:

:^nol

=

sinL

+

[

sin+

sk+

l+...

+

[

sin (k

++;

-sinG

+ )al

=sin(k*|t"

t+i;lr

sin (k +

+

)

ft - x) _dt

filrJ

= t

-x,iaJ=t'ti{

't

lnn'

'

lsin(

I'.!

1

.

z-x

sin(k+It,

s*(x)=;

_Ift'+x)

₊

Ysrn 1

I,1

.l'<ii -r-iL Yn ./rLs otlr

_{! ,-:9Vf(x).6y'}

srnfk+*)s

ds

s*(x) ₌

*

['f6

+ x)

sin(k+])t

^+t ? ..:-,,1 I

f

yr,lK,-y-:,j

f

++e

d(,-y : J -r

t

+ J -r

t

u-l

Ysm

-

Y

s*(x):

+

_I"ttrl

ds Ysrn

I

Y

(27)

*

!,"

1,*t-f

f-^)

.rli

-r-l.l

itj..f

.Jt? q ds ₌

L

_I"

t+

.j,

cosn sJds=\ (\ )

g-t,l

tffi

l.rsr(x)

Ju

s*(x)=+

!1r*r1

"'

c+

*

r'

ds

+*

Ift

*n

'i

t

*

_*'

_'r,

tu\/ n'-;\

r

ysrnl

fr-o'

ysrn-YY

sin(k+*)t

^['=I

['111x+s)-f(x-)+f(x-)J

_=a'"

ds -f fi ln'n. rr ' ttt " _vsrnl Y C*t-.r (\ ) q

.?i

t:

sin(k*|tt

rsur

I

Y

L

_I'fa-)

fr-n

r.

_=tlD-

+

\

'Yrt

i

_-Jl sin (k

*

{t

t

_ds-k)-r.I

_.f

Y

v)n

ds =

f(x')

sin(k*{t'

Ysllr Y cr-trtr vsrn

I

Y

*|t'

ds J Y o

I

-tt f(x+s) -

f(x-)

sin (k vsln

I

Y

6,,

f(x+s)-f!x-) = fim,f@+s)-f(x-)/

+

um _If@+s)'f(x')

t =f d-i

;;:-

ysrn

I

r{.-

s

ysrn

..{-

r*.-'

r

YY

5 1,31- r^,, _{d I}r _^",us r-l )1..:-, 1.,1 I -.-r al &s t

* rWC

b cr-lr

tr

Ysm

-

Y

rt; q+ t

tJ.",t5q

Jli,l

lim k+o

f(x+s)-f(x-)

_sin(k+I

_)sds=

o Ysln

I

'

Y Y

(28)

1+::t _J)

=

₊

r@+)

-a'*3

_Y

n=l

(an cosra

*

bn sin ru)

yhae

.''rlJt{

+ _f(x-)l

J|-r.i, .li q o+y ! -l *it--rr sx<rt d^eU

.trf6)

= x

*

tt

Cb

t's!

6f

'V

Jtb

.r.tl* b \ +

_+.

i.

...

cerd)r'

.Sr-5crli.lly.rr

_FL

t

:,F

x + xt

_{- {* f}

( J,. (- \),, cosra-

}

C

ln

sin

w)

f

_r=r

\

n\

n

i@)

=

tt

*

aY r,3-l

tto.i

t

.eot-"r x=lt

_t,

l.r

Cb

t':! 6r.rl&

iJFl

(r'

g:ltf(n+)='n+nl

, *

$

t1)n

cosr,.=

+

tf@+)+f(n-)l=

+

[@

+rt')+('n +,t')]

=,t'

..:-,1 _f.-^,_,IU

;

Yl ,-,11s or.ll tt _Cb

r"

AU

J'V

r'

+5

ddl

i,=''+.+*"'=!

'y*I'=ry

lim

r'

:

f(x+) k+o Y

lim.

s*(x) &'+o L+13 _JJ

(29)

)L.i l5

_{€e +j}

_t

g#,lf@)

_{A1'x ,,Li';.rt}l.r

Cb

4,.*rt

tSr

rt

q crrljl

..:,-,1- :L., .JL^

_,p

+t

t-lj

cSf _-2.:

gilLj

:f(-n)

=

f(n) J o),a

_[-n

,

n] i.,,lt.;)

r:-r.;r

_e.Uf(x)

+tS

utt .f

4..,ii

!

.rrir \f

@)

_Cb.rt

csr.Kji

.$u

13;h* $a'l3-t

{ru

_,t-l

if

_{@ ;r;a,.'a}

drr-f

jl

JirL-

tS*

S .tUT r.a-r.l

y,f(x)

u-t!

cSf ;l

rf.;

r, 4l^?

6$';t-g-.-,, trx

thi;y'

..1l

6,;i*-..>tt'-^

*

rt

Fil

+

_r&-)I

a. Y ,6 4.. \l -+LY _n=l

A.=+

$@)dx=*

tf@)-f(-n)t

=.

An=

\ ,t

f';lr

I7

g

cosra f,a

-

_|

_ffO

cosrul ltn+

-tt lt

I fO

-rg sinru dx

.r-iL

_J:

ott'l

t-.rrll

e|-rdbl) 4j

^,.f

:f €,; o.l .gLJl

+

>

(an cosru

*

bn sin nx)

tt= t

(30)

An=nbn; n=

\-lY-ll'-l

...

t^

=

_*

_{{O,

sin ru

tv

_-

_{} fO}

sin ruJ

; * [7o

cos ru dx

Bn=-flan;

n= \-lYr...

t ll.-

or.eli

jp

c.-;

*xf

_&)6

_*E

_rr

.t_id Jirt-

cs;t u-pi

Gr i

jtu- \t-,'l

_{il t}

6

;J*.1-+

,:,

(n

a,,sin

ru

.r nb,,cosru) =

+ lf

@+) +

f

(x-)]

.,,1 .Kl;i.r,.iL Yrr otlr

!.:-lL:.r

dta53 tc-,t;^t[-n ,

nl

tf@

#,-it.f

4-,ai

;t;1f&)

+,sl

_6r

t's1o

,Sf

.t-lr

*-t+/

(x)

6 _;blit.rr

_i'-lftf

f

(x) =

₊

*,,:,

(An cos

ta

+ Bn sin rc)

+i

C

i J;

Jsj,e

t1

@

f

(x)

-u;! 6f;l rl-,

q

d-+

_6S;*U.

S

(31)

(vt 1

... Jlt _J.ai

@

A

+

E

(a,, cosnx

*

b,, sin ru)

Y r=l

.tf

,s-$,t,S:t d-rti

_{f i {t?}

{,

d.?

JliC

"J*+U

-r,-!

6rlj;r

.gtJl

I - -'\-'

b

_'o+**f

IIo(ancosru*bnsinra)dvl

I rgf,a- { Y

,,=,

a f^^) ,-lL:; c-,,1 .,jtS ,l-l.''\ s

f ro, i

I

dx

-,t

+ [

@n sin nb - sin na) - b,, (cosnb

-

cos na)l

r+r,,/

)l; *

c--i.j

F(x)=

{

tfq-lto,

,i f

o;)tc

_S:t$l.1,-

^:''t-,F.:*,1 .SlnSs

:,f

r,(.!-l +. tb;

F'(x)=f(4-?

,.

=

_{] I"}

_lt f@) dx

llnotlr

!

,=sdfr(!-l .,;+j!

Jr-.c*,1

dl..*

uf,

-n

f--rl'r

F(x+yn)=

_{( rt,t-\}

_rc,

*

t'iiit

_I

_to'-F(x).

(32)

.-r-:!f

call*

catS.j:5

cSbl; r5.:*,1

ln

i22stf .f.lt

-

9

u7-t

,-;1.VF d/-lJt

l

F(x) (An cos nx

*

8,, sin ru)

F(x) b

!

rvdt={

a f(,) d, -

{

f0

dt .6

=L+I

_Y tt=r

=+

*:,

t

₊

(a,, sin

ra'

i

rat

dt

-

?'**

*

i

,o,*=

_?

@-a)*":,

i

,o,=?@'a),:,

_*,o^

cr-rr:u: rBn =

_ff

/,

= -

_*,.ri.,r

e5 bn cos ru)

l

6

..J

C

qr3 F(x)

-i,*f

.,

tf

; L

@,, sin nx - bn cos ra)

,:t

n

{<r-l

_{\ -b}

t-!

_{1a,sin nb-b,rcos}

nU-2

n

^=r

f;b

L

gsin

na-bncos na) n

o)i;Crir."

Jligtg^J-

O{l iJ.L,

,t'-lfd.:lp

e,

(sin nb

-

sin na) - bn _@osnb - cos na)J

(33)

(lr)

""JrlJ"i

!l.rr

\

f(x)

=

siru;

o1x1trt Cb q

Pv

,-t-*s

+';!

tsr

y\

JtI

.+k

lscos,

_db

qu;.r

_tSr

_di;l 6rKl;l' bn= o 1l - f(x)cosru d* =

L

fl

sinxcog&=

o

[n

,, x cosnx fl;6

*

--

v/ t + (-

])'/ ;

n*

\ n( \-n')

an=L

_I

IE

c,

=

Jl-

I

,t sinx = -lL +

jl- t

,t

,t O Asr cosx

_=--

a

fi n-, cos lrrJ

\

-

frl-n sin

lru

\-frl-

' c 1X1trt

filr

.:-,1

P tt

6,Wt c;ljl q an tS:'-l n

_-b

;

" 1x1rt

1'*t-f

,S

!

c*!tr

JeV

t7r :l

S-3rt.$l

tr-r -r.rt* l.2f$) =

x

; x

₌

y

_I

sinx -

ry

*

tin:'

- ...

_I

YT

|

[-cou.ry

#

+...]+c

i*

=,

[o"*

-Jt<x<lt

t;! tSf

.1

JU,

.J.tk trtt _Cb+,-s!

tSf

r*tr4s;Jtfst\

x' Y fi

t

-fi

(34)

'lr ? t+,i;l-r d/-lJq

.r'=

_$.

t,i +

cosra

C6.lt

t-.l39

_cSf

.Y.\

tp_,-t*iLr Yrr

.7rb.tlr

_Q

_{J.7jL;. *L.lr ! c-i}

_f(x,y)J+-f

rir-;;11.q.g"i

l-iLrl,.-r

jt5o;l"ul

t

_Clrs f@+yn,y)

=

f(x,y

+

fn) =

f(x,y)

C*t-.f

r

1c.*;f@/)

gb +'s!

,Sf

:,*,

[tt,n

0)

cosr/N

*

b,n g) sinmrJ

..li;r

5

aur9)

I

_f@,y)cosmx dx

h,,(!) ₌

* !"

f4,tl

sin mx rtx

1.,, tgi _J

rr-^

y.lt

-

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d/-lJt+ .:-.,1

=st;-y

u

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&:,-l

\

rbJ

rl.r F_L.r q's!

tSf

otf

+

JliC

i

(oru, cos ny

*

b,,r, sin ny)

flx,y)

_

a.(y)

*

Y

-\

xt @ -orn., SL

Y ,:r

arr(!)

(35)

(Yo)

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_J''i

b^(y) (c^n cos ny + d,*, sin ny)

@ -C^., LlF Y _n=l ,.li _-2.t

6

( _tal o,,* =

*

!*"

b,*,=

*

!,"

I

\ cmn-1 JT'

d^r=)

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f(x,y):?*+,:,

€ z=l

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cosnydxdy "

flty)

cos mx sin ny dx dy

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_{sin m}_x _cos_ny_dt_dy ( _tu1 ( _tcl (Y) tt tt

! I

ftxyl

sin mx -fl -Jl sin ny dt dy

[a.ncos ny+b.usin

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( _\d)

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3

|o,n.ro, m x*c,n;in rruvl

€

+X

m=l [a,*rcos mx

cos ny + b*ocos mx sin ny

+

crrrrsin mx cosny + dr*rsin mx sin nyJ

(36)

,itea,*,

-t

\--l_e.-o.t*i/(x,

-il

=

f6,il

:f?qfl

=

f&y).+lr?

d/-lJtl f(x,y)=

?

arr =)

_,t' f(x,y)

-?-,o*tot

mr cos

p|

(t)

o6y'

frlr.ti

(0) ..li _1r15 (v) .li;.: rS (^) = -f(x,y)

_+tr"

.rL-!.,,r

.

₊

2,o*'o'nY++i,

am'osmx*:,

1g 1t

I I

ft+yl

cos

mx

cos ny

&

dy

f(x,-y) =

-f&y),

f(-x,y) = f(x,y)

@@ b.,rsin ny +

2

ln=l ,l= I f(+y) = \ Y @@@ tn=l tn=l tt=l cos

nu

sin ny dx dy

f"*rlr ,G;i f(r,-y) =

f(+y)

,

f(xil

= -f(x,y)

_€

\s

Y _tt=r brr* ₌

* ['

!"f$,y)

(?)

fr

f

urut - a_7t'

I"n

I."fltr)

sin

mx

cos ny fu dy

(37)

(Yv) ..."..r'"'' JJI _J''l (q)

.li;r

+5 f(x,y)= (\ . ₎

=

t/

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.\

JU,

.J-rr V.p

_-rt11rt

+(!-l u. _th;

f(x,-y)

-

-ry =

-f(x,y),

f(-x,y) = -ry = -f(x,y)

f-.rlr

1",

y sin mxsin ny dx dY

=

(-\)'"*"

**

66 f(x,y)=

t

tn=I il=! ;l c..-,1

&;t* .b ,t-f

+'s-f

6f

_d/-lJt+ (-\)^*n

'

sin mx sin nY mn € m=l € ll= I

d-rti

_2r

ctS;r

u-t-t'

_6f

Js:a

₊l:f&Y)

drr. =

)

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t'fi'<x<rt

arr=

!

u-tPJltsJl .r.\

oif

f, Cb

-f I

rr-r;9

6f

gl.rlr

.:rL:'

Cb

r^

J^! cJ'rLjr

\'\

u:''".rr

ait+

.,,lt

_{- rf}

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+'t-tt

eV-f

t5Ll) _,r.rb,'.-.,A,r*;

y

=

f(x) ,' -6(x(o

(38)

.F-L.J

JfFl $.>,p' lolj'll,

tglu-2g

tSfdr:i"

J> J.,_r

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CttJj

U-l ,Sll..:-,,1

_lyy

+*t-f

JlFl

r,r5rlr

fi(x)=f(x)

;

-l<r<l

; p=\l

I

^S

;itsft@

eV -t-

r5.:*,lrrg-{

t}e

-b

,tl.; c^.,,1 olgJ.r:re 5,_/.1i.;r 15

.:-,,1 us-f

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_{L1lj:f(x) gb}.c,.."ly

: f(x) ;

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*

Cb JSJii, c--t+.r q j)..J-c,-,1

tlll*o

6 ,-?t

q':g

6f

_d/l

J-

_\f(x)

I

=timt?+

1o,,ro,

S1

*

bn sin

(

x))

l.+o Y

('

I O n= (x) , "

ijl

.t<ji rr-,1 c..:ti _{.trrr

M

Ji.rr

r5.[*

_lf@)ldx

-

M

<

*

_,-p-.r-!b

ti^ to.t=y:|

ttlatdxt=t:::+

{,'tro,td*

=t:,i

1irn,w*--tnU

=.

rf:r

_{Ltt, f(x)}

_{€ *b}

qJq

.z*,tf1g)

_Cb.r,

_tSr

jt

u*st

6r

f-rb,:*lrt+

6 f(x)

-

tim { l*o tt= t (ancos

+,

*

bn sin

ff

fll

it iy

drb.l.t

Jlrf.sl

otf

t;11b,, _5any-tlL _5wn

=

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Jb

,

,l.'\CJ

f=ot.r

*

sinw,,x

I

_ffq

sinwntdtl\ -t @ n=I

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-,,y+t

I [cosw,,x

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fO

cosw,,tdt -t p+r

rli

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wn+t-wn f(x) ,t I

c*t-r

Aw x

₊

,t;.rr

i;b ?-t'\

.6"i

,"i,

u^

n*

[,ftt)

cosw,,rdt

*

sinw,p

{,ial

sinw,,t

dt]

Lw}

=Lu^

_fr

l**

=*i

a(w) _f(x)coswxdx b(w) _f(x)sinwxdx f(x) [a(w)cos w x

*

h(w)sinw x

]

d w

lxl<o

lxl>n

(\a)

t

( \b)

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v-l.r- o

rj

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-\

i

'a -oo

f;

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r,

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t{ _biri 6c..,1

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_{Lfr 4 ii Lfr}

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_,Fl:

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f(x) €

[coswxIfOcoswtdt

-o

*

sr,n w x

_!

_fO

sin wt dtJ dw f(x) =

Cb.-.rt

JtFt

.\

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(40)

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Oye

912 1-;b: J'> @11

I lf@ldx*tI

xdx=n'

-@o

- \

itelcoswxdx=

l-

!"

7t _;- Jt b(w)=

_*

_IVelsinw

'

d'=

+

{p*rlr

j.hll

e.-:.-,1

t:+y

)-r1,1Cb.;.;r

JlFl

d/-lJtr a(w) xcoswxdx,= _" lt ["xsinw x dx=

]

t-

L

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f(x)

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f

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n

cos wn) sinw x d w

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ci,! _^,sg

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q

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t

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-

06y'

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_tf$)

Cb

.;t JlFl

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I-

to(w) cos wx

*

b(w) sinwx) dw =

|f

f@+) + f(x-)l

u-:gJlFrl

_s

_{cgr -hLi,}

esg*

.f

.\

*Jq+

Jli,-lrxiq

-

cos

|x.i

sin 0

sV-rt;l

ortn:-,'l U

cos

nr

=

I

_{Gtu +}

dtu),

skro

=

*

@tu - r'bo)

YYT

./-lL:.

_eb.--rt

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y-:Ur _;r-l.1r.,5;;(1t+ tr -l

y=f(x);

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_{; p=ltt}

i,f-f(x)

=

at-

a

2,

,o,

cosra

*

bn sinru)

{.*t-"r

f(x)

=

a.

a

i,, i

ro,'

ihn) ebo +

i

ro^ +

ib)'-iu

1

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f+)"/

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-

c. + ll=l (\ ₎

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,rr=*

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dn ₌

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tn1rl

,*ro d* .li.2r

d

fi

I

ro

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t

(YD) tt

I tg

@os ra - i sinru) iss

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ln

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6r.5..ru

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-,U.t

bl

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f$)

=

d .

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;

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=

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6fJ.Li,.,

t>2j-o

.\

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stt'

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q.rl

Gr

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rtt'

li

dJq

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+rit-cn =

+

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atu dx=

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ty

-+. d,

,(t'itt)x

l:"

,"=

_*

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##

yr(vin)r-r'(vin)n 1 =

ryr #

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filr

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t

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irf

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wr

: I

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r'i*)

;

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i

rtt*

+ e'i*)

(44)

f(x)

=

I*

t(

a(w) '-ib(w) )

,i*

+

Y,

f(x)

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[*

,@) ein' cl*

a(w) + ib(w)

) e'"* J dw

d(w) (a(w) + ib(w))

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c(w) =' - ib(w))

f(x) ( c(w)

,i*

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rl.:

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-\

Y @

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Irout

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l.

e\c

(q) (\ _"₎ (\ \ ) c(w) =

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e'i* dx _{; d(w)}₌

+

[_"fO

d*

dx c(w)

w

rri

-!-

f(')

e-'* dx c(w)

=

W _l_ IW) e'-^

(a

(\Y)

r:-,1

_lyy

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--* t

(\Y) J l"Li,.,

o2ye

+f(x)

u:!

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(\

\)

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;,*trt+ d(w) = c(-w)

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(w))

(\r)

c(w) (a(w)-ih(w));c(-w)

_-\

Y

-\

Y

{:

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_-

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L.2ttr,SJr

;

lrl

<n

;

lrl>n

f(x)

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"'

JrlJ"r

d/lJtl

.-t;s

w *rtt'qbJi

u-stJlFl

Ji.S.Str.rat*b

c(w) =

₊

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d* =

₊

_{7o

/*

d*

.

lo

=

_lwtt

i ,-i*l'- -

stn wlt |

-n

wir sln

wfi ,

,t dlt9 =

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f(')

=*{_

ry

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o;ye.r)-SlJ4

dl,

..:.-,1

ii

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a(w)=c(w)+c(-w)=rry

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.

f.r.ii .,1)

f(x)=+frycoswxdw

l1;1."'){:,r,:y

r*,,f-.r

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tl

_itoliC

*9

JlF"l

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1',,*llr

r6;ix

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rli

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r

4W=\

w

5r-5.+li

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_{-P F \:}

Y,

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lto O

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g*sinxdx=L

:xy

u-l9.r!,-t l

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ri \f(x)

eb

.rt

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c-fu

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r5

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srr/

l+:i

_bf@_Cttr.f

F"{fI

=

_e

_{I* fttl}

coswxdx

F" ifl

=

{1 f

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V}

--

i,

@) =

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f*

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_(r)

F',

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=

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ff)

.\

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I

o <X<O

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f/w)

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t 'tlLif

F,

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r

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ff6)

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+

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g)sin wx dr ₌

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rt.*t,-,coswx dxJ

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il.l

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r-J--r*l-

rt-'y",5.i\'ry

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d,

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--

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F

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rffit

6

J

f@) e''w dx -OO

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(

rt )

...

J5t.f'ai F {e'*'Y =,

: j_,

_,lln

F {e-*t'S =' F{af -1{ax +

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i,

I

+k

\f@) =

e'*' ;

a>.

gb

t-tg.Jrq

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*r'{*'*ir*) 4t \ {Yn

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J e

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"to'

l

dx T -w la T w

I

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@

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du --

G

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qi

tr = aF

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+ bF {sl

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S

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c-',J.tx

tF lf@ dg eS

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F

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c-;f

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r{gt

.t<ii r-iL s

sf

,-t!.i$-ry

qj

qr{g}

_{tFVI q.rt"}

.aiJ+q.=t

_fi

t .-at

_;l

jli:

_u._U

g;r;,1

..:,-,1#

_(f_._g)

t:!

,.-ptt-:J.-u

_-f

_tcsl^*

_{+*,sg$) rf@}

_{+s,rt.(.',t.t',}

_{d:"at)f 4-at}

F

_{f

rs}

=

,fr;

F{n

Fkt

FV.B)

=# f

_fn,ls@-t)s-i,,,xildx

I*

tTtrl

s6 - t) s-i',, dx dt .t<ji r.- lLr

tF I

_.'4l-3s;t'tiilo,,

_s

_tlx-f

..)LJf

\

--

_{rn

(51)

(

tt

_{Jrl J,..i}

FV.8)

=

l-r

_{,/ln -J}

I

.tft,1 g@) e-in(t + u) dt du g1u) e"i* du

=

_#

I-nq

e'i* dr

=ffi

FV:.

r

kl

C*t-.-r

6y

rss

-;

#-h:lrs$,-,'-€*,,-1,-r+

sE: p

t1t

,i:

FVI

Fiu.

@

(f .

g

_$)

=

_{f_fitr'lr}_E@)

,i*

d*

..:*,1

jf

.itc^:-

_!.tYrt-,

_JiL-,

_,F

elt

gV J-r;6

.t>J'l

{jbl ,-,t5Cr_l _.SkllJ)

_Cli

jl

dt

+,;g.i)t_"r; _,gly.

}s+

0J.,3

J- Jit-,

.t.\

.+1".t lt _-tj-;|.:.p,

t

_Cbu--g

6f

.\

Jtl,

O

I

(52)

lLn*,

;

-,t(r(c

f(x)-ll

1:n-X

i

o<X<rt

'Y

Jort

=

o \1.-,1

rt

Clt 1.3_l u.

j:

bn=L I"rinnxdx-

Y

pos!-cosrut)

7V

'n

tuft

Y

;

OrlJti

f(x)-L >

I1co,

nI

-cosnn)sinru r, Jt ,r:l n' Y

Cb.-rt

6r

.Y

Jt:,

+ttb

o^=* (l

f

_*"

+x)cosnxdx+

!."riz-x)cosnxdx

o,

=

*

V

₊

sin ru

.

ry

+

4

cos ruJ.o'

*

_f

_+sinru

_{ry-)cosral.nl}

=fif\

+

(-\)"*')

o.

-+

_r

I,+"

+x)dx

+

li+-odrt

=

.

f:lr

.:,.,1

_6rj

_6bO.;a bn= o etl.;atti

f&)=+

_1l

f *

k\)n*,+\)costu

,r: r

(53)

.+tr

lrf(x) = x + si^nx

,

-trt<x<lt _Cb

t t 6r 'f

J[le

f(-x)=-f(x)

i

an=

o

b,=

₊

!."@ + siru)sinradx =

fr f

_!1

t*nxdx

+

["ri*sinrudx]

b,=r+jl-

_7to

fnrin'rdx=Y

crrtrQ

f(x)

-Ysinr

+

Y; +

_n=l n sinru cb

*rt tsr

.f Jtb

(fr)

....

Jrl J.ei

; p=l

.+tt* b

an

:

!'

rro, nn x dx

+

_J'(r

-

x)

cos n

tafla

- #

kV"

-

l)

a.

3 o,

bn

=

!'

r

rin nn xdx

+

J'(r

- x) sin nn xdx=

₊

kV"*'

+

\) f(x)

=

i

,

#

(-\f'-

r,

*r)+

J

('v"*'-

\) sinnnxl n=l

('

_''

ocx<\ f(x)

=

{

Lt-t ;

\<xcY

(54)

.+t

t

ltf&)

-

sf

;

-*1r1o

_{Cb t-t-f}

6r

.0

Jtl,

f, e- cos nx ax

_{- \ ,}

dsin nx

;t

"

s' sin

ru l"

1 ,

-e'cos nx

" i-"-;t

"

",

+

1"t

'u^*1

":

+ InT'o'*

'*l

.tt

ott: t

J

7t -il

ln .

n

| +

t

Je*cosnxtult

l-"

n

-.t

-J-{

xt sinh n lt d/lJt+

f(x)

*

3 +({

@os ra - n sin

ra)l

,r=! \ *n'

(55)

(to)

_"''Jrl

Ji''i ,sT r,.,.+7;trrr.:t"r

\f&)

=.rt

,

-l<x</ _Cbt'-s!

6f

't4-

v))te

,?

Jlb

..r*.rric--,)1b \

.

+

.

+

+... r \ -

+

.

+-+...

G)-v

s!-ril4.

-p

t-)w

o^=*

!'r'ro,ffxdx=+

r*

!',,uT*t

- r t

,ror

y l'=

_I'',

(-\)n

I

,to

tut tutt

,

_lo

flfr

a.=+{tt,d*=+

f(x) =

x'

=

₊

.

#,i

)

r- \)n cos

+,

,.*t-fx

=ntl=n

f-r)

it*d.lt;,..r-l

t: ft,SV

n'=o'

* i

il-r.;,

n'

'+.++*...=l

,_ \ + \ _ \

t-T-F

+

_ :o'

-"-

\Y

f;l:x

=

o

tl

₌

n

,*^)

)lj Jl

JV

.+tt

\tf@

= sinnx

i

o(r(

\,

P

=

\

_Cb

+s!Gr.V

Jt:"

ll

an: I

_I

sinnxcoslwcxdx=

_I

_[tU(ln

*

\)ntx-sin(ln-

l)nx]

fu

(56)

-Y\

-X-It fn'- \

-Y

_1t f(x)

-Y_f

_It

_rt

f

I

cosnnx ,r=t Ynt- \ .r.rtt

_\f(x)

=

x'

,'

_{-Y<.r<Y Cb q':g}

6f

..^

Jt!

Jan= o I

br= [

x'sin"" xdx=[

?.nfi Y

Yxr

_cos-l

,ot

l'

run

Y

_l.

+

l--

!.'r, ,o,

T

*,

-

\t(-\)"+'

_fun

+ ? I_ I f"rinnnx dxl=

_ntrtr

_'

_rut

-o

_{Y '}

\2(-\)n+'farc

bn _--(-

\)"*'

\?n'n'-

qt

n'trt'

f(x) \?n'n' _n'trtt- \?

.r-tf* U

f@

=

e'','

o(x(Y't

_Ctt+'-s!

6r.q

JU'

ttl

e-' cos

w

dx =

*

[

e't ('costu

+

nsin

w].'o

n(n' +

\)

_' q2(- _\)"+' ntfit d/-lJttl

.

nfix sut

-

Y 6

s

L. n=I

an=I1

,a (-

\)"*'

(57)

(rv)

Jrl J.ai

=1\-e'tn )+

_'

_fi

_n'+\

: a.=\'e''o

_rt

bn =

₊

!'n

ro rin

w

d^x, =

#

[

e-' (-sin ru

*

n

cos ta)],'o

- \-t'to

_rt

. ,,,

_nt+\

d/-lJLt

e-,

= \-e''o

t

(I+ I

cosx++

coslx+I

cosyx+...) +

(Is;ru

+{

fi

, y y-- .O'""'-

\o

,y

0 sintx +

_-!-

sinYx + ...) )

\"

CV

g-r+r,*rJ

cgr

.\

"

Jtb

.+k

b bn

-

r,{

_f@

sin nn x dx

=r[

{',*

-x)sinnnxdx+

!,' O-{f

_T sinnnxdv)

[I-,

,

"o.i

(58)

- \

cos nfix

=r[-(=-x)-

_Y

_tuft

ffii+rt'$-irry

*+ant

=

fi

t\+(-r)n/

n*Y Sln fUfX _,r

TI,

_T

=#t\-(\rt

-W

atfttr

I@) = (

_+-

]t'i,,'**

(*

.

#

)

siny'o*

(*-

#

)

sinutx

*...

.+k

ttf@

= srnfr ; o 1x1z Cb

.friJ

u_sf

6f

.\

\

Jtlr

o^ =

_*

_fnrinr,

cos ,x d.r

-

_*

_,

_Y

[;

!.nr^yx sin nx dx J

-?

_,

'cosYx cos

nx

l"

-,*I--

I. I

on=fu(t+1-1'1',l

;

or=oi o.=*

fk)

=-L-*

f

? e' _'\

'

Yn

'nj,

_,

_{,*, fu}

(\ +(- \)") cos ru

.LtSi .t-.r: u.ftx) = _{coslu t-lt<x<o}_Cb

t;! 6flab,

:-t22,a.lI

J['lr

+

_4on

n'

crr =

+ I"

costu ,'ittr fla

-

{ t

e-bo

sinta

(59)

(ff )

...

Jrl _Jt.i

=

(- L)" sinh

n

+

[

e

-bo cos lr;l"

*,

!

or _-ro

cosla dxJ

,t

llt'

_{l_"}

:n

=

(-\)n_ sinhn + in(-l)n cosh n 'ntcn

It ;

-tt(.r(o f(x)

=

{

[cosr

;

.1x1nr

cn=#sinhn.#incoshn

f(x)

=*,i_

_frf,,

[sinhn + incoshn

_]e'bo

Cb.;y

GrlL;,

c.,.tt'

.\f

JIL

.r.ttd _b

cn ₌

_*

7"

r,r)

e-i'o dx

- +

I"

r'uo dx

+

I.n rour'bo d,

=

e'bo l;

[.n@k

+e'o)r-budx

: 't _k'in,t-\)+ _{' [\} _riz(vn)- \ + \ ,'itt(t+n)- \ /

Ynn

ln 'n-\

n-\ n*\

n*\'

= i

(\_(-\)n)+

i t(-\)"'_ \ (-\)"*'_ \

r

(60)

=#(\-(-\)n)+#

#

|,*.,\

c,=I

f e'bdx+ \

lnroue-bdx=

l+J

_nf

ln :n

ln

-o n"ou dx

_-

\ Y \

L.--ln

{,'

r

Sr

_tt

I"d,

*

t

'tl o

f(x)-

+.

(L+Lv+ \ ;

_{' Jt} _f ln rt=_@ n*o,\ Y JT

f

JT

n r3

_Z/

Y

fr _n:, ; -fi(I(c ; " 1X1Tt

{Ly #

rcv"*,_\)rp

+;!

6r:te$t-3s;t

tt

.\f

Jt!

; -tt (-f,( o ; " 1X17t

3

sin(rn- t)x - Jl*t -n:

r

Ytt- \ ,

g-$c

_dr*,b

JlFl

*-b;l

l-r;,1

.*t

ltl

Cb u-sg

6r

+

3

sin(Yn- \)x

z-

_-!y61

ax:

n=r

Yn-\

S

cos(Yn-\)x

[-x

;

-nt(r(

o

;, -lIn-lf - l, ;

.1x11t

:.r-lrtr

, =

+

,.*t-.r -n l:n

{trti

.,,

il

g.sJ

d"hjl

.Jr$lFl

ti cos(ln-\)x

-[-x

;

-nt('t( o

-@-1r

i

o.)c1n