Function Theory E

8/12/2019 Function Theory E

1/19

KI\OU

"kibfsogukirpoysfes.fb" 9

ub tfmbs

L`hfbft fmb 4Hubetfmb fs i ruj` (mr emrr`spmbl`be`), hrmk i bmb `kpty s`t I tm i bmb `kpty s`t A, toit issmefit`s

ìeo k`ka`r mh I tm i ubfqu` k`ka`r mh A. Uykamjfeijjy, w` wrft` h4 I A. S` rìl ft is "h fs ihubetfmb hrmk I tm A".

Hmr `xikpj`, j`t I {9, >, 9} ibl A {>, 9, 1}.\o`b I A {(9, >), (9, 9), (9, 1), (>, >), (>, 9), (>, 1), (9, >), (9, 9), (9, 1)}Bmw, " h 4 I A l`hfb`l ay h(x) ; x1 " fs to` hubetfmb sueo toith {(9, 9), (>, >), (9, 9)}h eib ijsm a` somwb lficrikitfeijjy ay hmjjmwfbc kippfbc.

I A

Bmt` 4 @v`ry hubetfmb siy y ; h(x) 4 IA. O`r` x fs fbl`p`bl`bt virfiaj wofeo tig`s fts viju`s hrmk I wofj`'y' tig`s fts viju` hrmk A. I r`jitfmb wfjj a` i hubetfmb fh ibl mbjy fh

(f) x kust a` iaj tm tig` ìeo ibl `v`ry viju` mh I ibl

ibl (ff) mb` viju` mh x kust a` r`jit`l tm mb` ibl mbjy mb` viju` mh y fb s`t A.

Cripofeijjy 4 Fh iby v`rtfeij jfb` euts to` cripo it kmr` toib mb` pmfbt, to`b to` cripo lm`s bmt

r`pr`s`bt i hubetfmb.

@xikpj` # 9 4 (f) Sofeo mh to` hmjjmwfbc emrr spmbl`be`s eib a` eijj l i hubetfmb =

(I) h(x) ; x5 7 {9, >, 9} {>, 9, 1, 5}

(A) h(x) ; x 7 {>, 9, , 9, 1}

(E) h(x) ; x 7 {>, 9, , 9, 1}

(L) h(x) ; x 7 {>, 9, , 9, 1}(ff) Sofeo mh to` hmjjmwfbc pfetmrfij lficriks r`pr`s`bt to` hubetfmb

(I) (A)

(E) (L)

Umjutfmb 4

(f) h(x) fb (E) ibl (L) ir` hubetfmbs is l`hfbftfmb mh hubetfmb fs sitfshf l. wofj` fb eis` mh (I) to`

cfv`b r`jitfmb fs bmt i hubetfmb, is h(9) 1bl s`t. O`be` l`hfbftfmb mh hubetfmb fs bmt sitfshf`l.Sofj` fb eis` mh (A), to` cfv`b r`jitfmb fs bmt i hubetfmb, is h(9) ; 9 ibl h(


2/19

KI\OU


U`jh prietfe` prmaj`k 4

(9) J`t c(x) a` i hubetfmb l`hfb`l mb W9, 9[. Fh to` irì mh to` `qufjit`rij trfibcj wfto twm mh fts

v`rtfe`s it (>,>) ibl(x,c(x))fs 5

|x| 0 x 0 mr x 0

to` lmkifb mh h fs (, 0 [ W 0 , )

(ff) sfb9 (1x 9) fs rìj fhh 9 1x 9 + 9 lmkifb fs x W>, 9[

Ijc` arife M p`ritfmb s mb Hu betfm bs 4Fh h ibl c ir` rìj viju`l hubetfmbs mh x wfto lmkifb s`t I ibl A r`spètfv jy, to`b amto h ibl c ir`

l`hfb`l fb I A. Bmw w` l`hfb` h + c, h c, (h . c) ibl (h /c) is hmjjmws4

(fff)

c

h(x) ;

)x(c

)x(hlmkifb fs {xx II A sueo toit c(x) >}.

Bmt` 4 Hmr lmkifb mh(x) ; {h(x)}c(x) , embv`btfmbijjy, to` emblftfmbs ir` h(x) 2 > ibl c(x) kust a` rìj.


3/19

KI\OU


Hmr lmkifb mh(x) ; h(x)Ec(x)mr(x) ; h(x)^c(x)embv`btfmbij emblftfmbs mh lmkifb ir` h(x) c(x)ibl h(x) B ibl c(x) S

@xikpj` # 5 4 Hfbl to` lmkifb mh hmjjmwfbc hubetfmbs 4

(f) h(x) ; 1sfb x 98 x

(ff) h(x) ; 1

5

< x jmc(x5

x)

(fff) h(x) ; xems 9x

Umjutfmb 4 ( f) sfbx fs rìj fhh sfb x > xW1b, 1b + [, bF.

198 x fs rìj fhh 98 x 1 > < x . lmkifb mh jmc(x5 x) fs (9, > ) (9,).

O`be` to` lmkifb mh to` cfv`b hubetfmb fs {( 9, > ) (9,)}(1, 1) (9, > )(9, 1).

(fff) x 2 > ibl 9 x 9 lmkifb fs (>, 9[

U`jh prietfe` prmaj`ks 4

(5) Hfbl to` lmkifb mh hmjjmwfbc hubetfmbs.

(f) h(x) ;)x1jmc(

9

+ 9x (ff) h(x) ; x9 sfb9

5

9x1

Ibsw`rs 4 (f) W9, 9) (9, 1) (ff) W9, 9[

K`to mls mh l`t`rkfb fbc rib c` 4(f) Q`pr`s`btfbc x fb t`rks mh y

Fh y ; h(x), try tm `xpr`ss isx ; c(y), to`b lmkifb mh c(y) r`pr`s`bts pmssfaj` viju`s mh y, wofeofs ribc` mh h(x).

@xikpj` # < 4 Hfbl to` ribc` mh h(x) ;9xx

9xx1

1

Umjutfmb 4 h(x) ;9xx

9xx1

1

{x1 + x + 9 ibl x1 + x 9 oiv` bm emkkmb hietmr}

y ;9xx

9xx1

1

yx1 + yx y ; x1 + x + 9

(y 9) x1 + (y 9) x y 9 ; >Fh y ; 9, to`b to` iamv` `quitfmb r`lue`s tm 1 ; >. Sofeo fs bmt tru`.

Hurto`r fh y 9, to`b (y 9) x1 + (y 9) x y 9 ; > fs i quilritfe ibl ois rìj rmmts fh(y 9)1 < (y 9) (y 9) > f.`. fh y 5/0 mr y 9 aut y 9\ous to` ribc` fs (, 5/0[ (9,)

(ff) C ri po feij K `t om l 4\o` s`t mh y emmrlfbit`s mh to` cripo mh i hubetfmb fs to` ribc`.

@xikpj` # 0 4 Hfbl to` ribc` mh h(x) ;1x


4/19

KI\OU

"kibfsogukirpoysfes.fb" , 9[

Bmw, usfbc kmbmtmbmefty mhb t,

b (1x x1) (, >[ ribc` fs (, >[ Ibs.(ff) y ; s`e9 (x1 + 5x + 9)

J`t t ; x1 + 5x + 9 hmr x Q, to`b t

,, to`b h fs eijj`l ipmjybmkfij hubetfmb mh l`cr`` b.

Bmt` 4 \o`r` ir` mbjy twm pmjybmkfij hubetfmbs, sitfshyfbc to` r jitfmb7 h(x).h(9/x) ; h(x) + h(9/x), wofeo

ir` h(x) ; 9 xb


6/19

KI\OU


^rmmh 4 J`t h(x) ; i>xb + i9 x

b 9 + ......... + ib, to`b h

x

9; b

>

x

i+ 9b

9

x

i + ......... + ib.

Ufbe` to` r`jitfmb omjls hmr kiby viju`s mh x,

Emkpirfbc to` em`hhfef`bts mh xb, w` c`t i>ib; i> ib; 9

Ufkfjiry emkpirfbc to` em`hhfef`bts mh xb 9 , w` c`t i>ib 9 + i9ib; i9 ib 9 ; >,

jfg` wfs` ib 1, ......, i9 ir` ijj z`rm.

Emkpirfbc to` embstibt t`rks, w` c`t 1

b19

1>

i.......ii ; 1 1bi i> ; 9

(ff) I jc à ri fe H ub et fm b 4y fs ib ijcàrife hubetfmb mh x, fh ft fs i hubetfmb toit sitfshf s ib ijcàrife `quitfmb mh to` hmrk,

^> (x) yb + ^9 (x) y

b9 +....... + ^b9 (x) y + ^ b(x) ; >, wo`r b fs i pmsftfv` fbt`c`r ibl

^> (x), ^9 (x)....... ir` pmjybmkfijs fb x. `.c. y ;xfs ib ijcàrife hubetfmb, sfbe` ft sitfshf`sto` `quitfmb y x ; >.

Bmt` 4 Ijj pmjybmkfij hubetfmbs ir` ijcàrife aut bmt to` embv`rs`.

I hubetfmb toit fs bmt ijcàrife fs eijj`l\ribse`bl`btij Hubetfmb.

(fff) Q itf mb ij H ub et fm b 4

I ritfmbij hube tfmb fs i h ubetfmb mh to` hmrk, y ; h ( x) ;)x(o

)x(c, wo`r` c (x) ibl o (x), o(x) > ir`

pmjybmkfijs.

(fv) @ xpm b` bt fi j Hub et fm b 4I hube tfmb h(x) ; ix ; `x Fb i (i 2 >, i 9, x Q) fs eijj`l ib `xpmb`btfij hubetfmb. Cripo mh

`xpmb`btfij hubetfmb eib a` is hmjjmws 4Eis` - Eis` -Hmr i 2 9 Hmr > : i : 9

(v) Jmcirftokfe Hubetfmb 4 h(x) ; jmcix fs eijj`l jmcirftokfe hubetfmb, wo`r` i 2 > ibl i 9 iblx 2 >. Fts cripo eib a` is hmjjmws

Eis`- Eis`-Hmr i 2 9 Hmr > : i : 9


7/19

KI\OU


(vf) I as mj ut` X ij u` H ub et fm b / K ml uj us H ub etf mb 4\o` sykamj mh kmlujus hubetfmb fs h (x) ;xibl fs l`hfb`l is4 y ;x

>xfhx

>xfhx

(vff) Ufcbuk Hubet fmb 4 (Ijsm gbmwb is scb(x))I hubetfmb h (x) ; scb (x) fs l`hfb`l is hmjjmws 4

h (x) ; scb (x) ;

>xhmr9

>xhmr>

>xhmr9

Ft fs ijsm wrftt`b is scb x ;

>x7>

>x7

x

|x|

Bmt` 4 scb h(x) ;

>)x(h7>

>)x(h7)x(h

|)x(h|

(vfff) C rì t` st Fb t`c `r H ub et fm b mr U t` p H ube tf mb 4\o` hubetfmb y ; h (x) ; Wx[ fs eijj`l to` crìt`st fbt`c`r hubetfmb, wo`r` Wx[ `quijs tm to`

crìt`st fbt`c`r j`ss toib mr `quij tm x. Hmr `xikpj` 4

hmr 9 x : > 7 Wx[ ; 9 7 hmr > x : 9 7 Wx[ ; >hmr 9 x : 1 7 Wx[ ; 9 7 hmr 1 x : 5 7 Wx[ ; 1 ibl sm mb.

^ rm p`r tf `s m h c rì t` st f bt `c `r h ube tf mb 4(i) x9 : Wx[ x

(a) Fh k fs ib fbt`c`r, to`b Wx k[ ; Wx[ k.

(e) Wx[ + Wy[ Wx + y[ Wx[ + Wy[ + 9

(l) Wx[ + Wx[ ; `c`rfbtibbmtfsxfh,9

`c`rfbtibfsxfh,>

(fx) H ri et fm bi j ^ ir t Hub et fm b4Ft fs l`hfb`l is, y ; {x} ; xWx[, wo`r` W.[ l`bmt`s crìt`st fbt`c`r hubetfmb.


8/19

KI\OU

"kibfsogukirpoysfes.fb" ?

`.c. to` hrietfmbij pirt mh to` buka`r 1.9 fs 1.9 1 ; >.9 ibl {5.3} ; >.5.\o` p`rfml mh tofs hubetfmb fs 9 ibl cripo mh tofs hubetfmb fs is somwb.

(x) l `b tf ty hub et fm b 4\o` hubetfmb h 4 I I l`hfb`l ay, h(x) ; x,xI fs eijj`l to`fl`btfty hubetfmb mb I ibl fs l`bmt`l ayI. Ft fs ìsy tm mas`rv`toit fl`btfty hubetfmb fs i afdètfmb.

(xf) E mb st ib t h ub et fm b 4I hubet fmb h 4 I A fs sifl tm a` i embstibt hubetfmb, fh `v`ry`j`k`bt mh I ois to` sik` h fkic` fb A. \ous h 4 I A7h(x) ; e, x I, e A fs i embstibt hubetfmb.

@xikpj` # ? 4 (f) J`t {x} ibl Wx[ l`bmt to` hrietfmbij ibl fbt`crij pirt mh i rìj buka`r x r`spètfv jy.

Umjv` > >

95

1

5

0

\o`r` ir` twm smjutfmb mh cfv`b `quitfmb x ; > ibl x ;5

0

(ff)


(0) Fh h 4 Q Q sitfshyfbc to` emblftfmbs h(>) ; 9, h(9) ; 1 ibl h(x + 1) ; 1h (x) + h(x + 9), to`b hfblh (8).

(8) Lriw to` cripo mh hmjjmwfbc hubetfmbs, wo`r` W.[ l`bmt`s crìt`st fbt`c`r hubetfmb

(f) y ; W 1 x [ + 9 (ff) y ; x W x[, 9 x 5 (fff) y ; scb (x1 x)

Ibsw`rs 4 (0) 8


9/19

KI\OU


(8) (f) (ff) (fff)

@q uij mr l`btfei j Hubetfmbs 4\wm hubetfmbs h ibl c ir` sifl tm a` fl`btfeij (mr `quij) fhh 4

(f) \o` lmkifb mh h to` lmkifb mh c.

(ff) h(x) ; c(x), hmr v`ry x a`jmbcfbc tm to`fr emkkmb

lmkifb.

`.c. h(x) ;x

9ibl c(x) ; 1x

xir` fl`btfeij hubetfmbs.

Ejìrjy to` cripos mh h(x) ibl c(x) ir` `xietjy sik`

Aut h(x) ; x ibl c(x) ;x

x1

ir` bmt fl`btfeij hubetfmbs.

Ejìrjy to` cripos mh h(x) ibl c(x) ir` lfhh`r`bt it x ; >.

@xikpj` # 6 4 @xikfb` wo`to`r hmjjmwfbc pifr mh hubetfmbs ir` fl`btfeij mr bmt =

(f) h(x) ;9x

9x1

ibl c(x) ; x + 9

(ff) h(x) ; sfb1x + ems1x ibl c(x) ; sè1x tib1x

Umjutfmb 4 (f) Bm, is lmkifb mh h(x) fs Q {9}

wofj` lmkifb mh c(x) fs Q

(ff) Bm, is lmkifb ir` bmt sik`. Lmkifb mh h(x) fs Q

wofj` toit mh c(x) fs Q

Fb7

19b1

U`jh prietfe` prmaj ks

(3) @xikfb` wo to`r to` hmjjmwfbc pifr mh hubetfmbs ir` fl`btfeij mr bmt 4

(f) h(x) ; scb (x) ibl c(x) ;

>x>

>x|x|

x

(ff) h(x) ; sfb9x + ems9x ibl c(x) ;1

Ibsw`rs 4 (f) V`s (ff) Bm


10/19

KI\OU

"kibfsogukirpoysfes.fb" 9>

Emkpmsf t` Hubetfmb 4J`t h4 _V

9ibl c4 V

1 T a` twm hubetfmbs ibl L fs to` s`t mh viju`s mh x sueo toit fh x_, to`b h(x) V

1.

Fh L , to`b to` hubetfmb o l`hfb`l mb L ay o(x) ; c{h(x)} fs eijj`l emkpmsft` hubetfmb mh c ibl h ibl fsl`bmt`l ay cmh. Ft fs ijsm eijj l hubetfmb mh i hubetfmb.

Bmt` 4 Lmkifb mh cmh fs L wofeo fs i suas`t mh _ (to` lmkifb mh h ). Qibc` mh cmh fs i suas`t mh to` ribc` mh

c. Fh L ; _, to`b h(_) V1.

^fetmrfijjy cmh(x) eib a` vf`w l is ubl`r

Bmt` toit cmh(x) `xfsts mbjy hmr toms` x wo`b ribc` mh h(x) fs i suas`t mh lmkifb mh c(x).

^rmp`rtf`s mh Emkpmsft` Hubetfmbs 4(i) Fb c`b r ij cmh hmc (f.`. bmt emkkutitfv`)(a) \o` emkpmsftfmb mh hubetfmbs ir` issmefitfv` f.`. fh tor`` hubetfmbs h, c, o ir` sueo toit

hm (cmo) ibl (hmc) mo ir` l`hfb`l, to`b hm (cmo) ; (hmc) mo.

@xikpj` # 9> 4L`serfa` hmc ibl cmh wo`r`v`r fs pmssfaj` hmr to` hmjjmwfbc hubetfmbs

(f) h(x) ; x 5 , c(x) ; 9 + x1 (ff) h(x) ; x , c( x) ; x1 9.

Umjutfmb 4 (f) Lmkifb mh h fs W5,), ribc` mh h fs W>,).Lmkifb mh c fs Q, ribc` mh c fs W9,).

Hmr cmh(x)

Ufbe` ribc` mh h fs i suas`t mh lmkifb mh c,

lmkifb mh cmh fs W5,) {`quij tm to` lmkifb mh h }

cmh (x) ; c{h(x)} ; c ( x 5 ) ; 9 + (x+5) ; x + ,).Lmkifb mh c fs Q, ribc` mh c fs W9,).

Hmr cmh(x)

Ufbe` ribc` mh h fs i suas`t mh to` lmkifb mh c,

lmkifb mh cmh fs W>,) ibl c{h(x)}; c(x) ; x 9. Qibc` mh cmh fs W9,)Hmr hmc(x)

Ufbe` ribc` mh c fs bmt i suas`t mh to` lmkifb mh h

f.`. W9,) W>,) hmc fs bmt l`hfb`l mb womj` mh to` lmkifb mh c.

Lmkifb mh hmc fs {xQ, to` lmkifb mh c 4 c(x)W>,), to` lmkifb mh h}.\ous to` lmkifb mh hmc fs L ; {x Q4 > c(x) :}

f.`. L ; { xQ4 > x 1 9}; { xQ4 x 9 mr x 9 }; (,9[ W9,)

hmc (x) ; h{c(x)} ; h(x 19) ; 1x 9 Fts ribc` fs W>,).

@xikpj` # 99 4 J`t h(x) ; `x 7 Q+ Q ibl c(x) ; sfb9 x7 W9, 9[

1,

1 . Hfbl lmkifb ibl ribc` mh hmc (x)

Umjutfmb 4 Lmkifb mh h(x) 4 (>,) Qibc` mh c(x) 4

1

,

1

viju`s fb ribc` mh c(x) wofeo ir` iee`pt`l ay h(x) ir`

1,>


11/19

KI\OU


> : c(x)1

> : sfb9x

1

> : x 9

O`be` lmkifb mh hmc(x) fs x (>, 9[

\o`r`hmr` Lmkifb 4 (>, 9[

Qibc` 4 (9, /1[

@xikpj` # 91 4Emkpmsftfmb mh pfè`wfs` l`hfb`l hubetfmbs 4Fh h(x) ; | |x 5| 1 | > x x ) (9,) Lmkifb 4 (>,)Qibc` 4 (9,) Qibc` 4 (>,)

Ejiss fh feit fmb mh Hubetfmbs 4Hubetfmbs eib a` ejissfhf`l is "Mb` Mb` Hubetfmb (Fbdètfv` Kippfbc)" ibl "KibyMb` Hubetfmb"4

Mb` Mb` Hubetfmb 4I hubet fmb h 4 I A fs sifl tm a` i mb`-mb` hubetfmb mr fbdètfv` kippfbc fh lfhh`r`bt `j`k`bts mh I oiv`lfhh`r`bt h fkic`s fb A.

\ous hmr x9, x1 I ibl h(x9), h(x1) A, h(x9) ; h(x1) x9; x1mr x9 x1 h(x9) h (x1).Lficrikkitfeijjy ib fbdètfv` kippfbc eib a` somwb is

MQ

Kiby Mb` hubetfmb 4I hubetf mb h 4 IA fs sifl tm a` i kiby mb` hubetfmb fh to`r` `xfst it jìst twm mr kmr` `j`k`bts mh Ioivfbc to` sik` h fkic` fb A.

\ous h 4 I A fs kiby mb` fhh to`r` `xfst itjìst twm `j`k`bts x 9, x1 I, sueo toit h(x9) ; h(x1) autx9 x 1.

Lficrikkitfeijjy i kiby mb` kippfbc eib a` somwb is

MQ


13/19

KI\OU


Bmt` 4 Fh i hubetfmb fs mb`mb`, ft eibbmt a` kibymb` ibl vfe` v`rsi.

K`tomls mh l`t`rkfbfbc wo`to`r i cfv`b hubetfmb fs MB@ M B@ mr KIBV M B@ 4(i) Fh x9, x1I ibl h(x9), h(x1)A, `quit` h(x9) ibl h(x1) ibl fh ft fkpjf`s toit x 9; x1, to`b ibl mbjy

to`b hubetfmb fs MB@-MB@ mto`rwfs` KIBV-MB@.

(a) Fh to`r` `xfsts i strifcot jfb` pirijj`j tm x-ixfs, wofeo euts to` cripo mh to` hubetfmb itjìst it

twm pmfbts, to`b to` hubetfmb fs KIBV-MB@, mto`rwfs` MB@-MB@.

(e) Fh `fto`r h (x) >, x lmkifb mr h(x) > x lmkifb, wo`r` `quijfty eib omjl it lfser`t`pmfbt(s) mbjy f.`. strfetjy kmbmtmbfe, to`b hubetfmb fs MB@-MB@, mto`rwfs` KIBV-MB@.

Bmt` 4 Fh h ibl c amto ir` mb`-mb`, to`b cmh ibl hmc wmujl ijsm a` mb`-mb` (fh to`y `xfst). Hubetfmbs eib ijsm

a` ejissfhf`l is "Mbtm hubetfmb (Uurdètfv` kippfbc)" ibl "Fbtm hubetfmb"4

Mb tm hub etfmb 4Fh to` hubetfmb h 4 I A fs sueo toit ìeo `j`k`bt fb A (em lmkifb) kust oiv` itjìst mb`pr`fkic` fb I, to`b w` siy toit h fs i hubetfmb mh I 'mbtm' A. \ous h 4 I A fs surdètfv` fhh a A,to`r` `xfsts smk` i I sueo toit h (i) ; a.

Lficrikkitfeijjy surdètfv` kippfbc eib a` somwb is

MQ

Fbtm hub etfmb 4Fh h 4 I A fs sueo toit to`r` `xfsts itjìst mb` `j`k`bt fb emlmkifb wofeo fs bmt to` fkic` mh iby`j`k`bt fb lmkifb, to`b h(x) fs fbtm.

Lficrikkitfeijjy fbtm hubetfmb eib a` somwb is

MQ

Bmt` 4 (f) Fh ribcèmlmkifb, to`b h(x) fs mbtm, mto`rwfs` fbtm

(ff) Fh i hubetfmb fs mbtm, ft eibbmt a` fbtm ibl vfe` v`rsi.

I hubetfmb eib a` mb` mh to`s` hmur typ`s4(i) mb`mb` mbtm (fbdètfv` ibl surdètfv`)

(a) mb`mb` fbtm (fbdètfv` aut bmt surdètfv`)

(e) kibymb` mbtm (surdètfv` aut bmt fbdètfv`)

(l) kibymb` fbtm (b`fto`r surdètfv` bmr fbdètfv`)

Bmt` 4 (f) Fh h fs amto fbd etfv` ibl surd etfv`, to`b ft fs eijj`l iafdètfv`kippfbc. \o` afdètfv` hubetfmbs

ir` ijsm bik`l is fbv`rtfaj`, bmb sfbcujir mr afubfhmrk hubetfmbs.


14/19

KI\OU

"kibfsogukirpoysfes.fb" 9, 5[I. Hfbl wo`to`r h(x) fs fbdètfv` mr bmt. Ijsm hfbl to` s`t I, fhh(x) fs surdètfv`.

Umjutfmb 4 (f) \o` lmkifb mh h(x) fs Q. h (x) ; 9 sfb x. h (x ) > x emkpj`t` lmkifb ibl `quijfty omjls it lfser`t` pmfbts mbjy h(x) fs strfetjy fberìsfbc mb Q. O`be` h(x) fs mb`-mb`.(ff) Is ribcèmlmkifb, to`r`hmr` cfv`b hubetfmb fs MB\M

(fff) h (x) ; 1(x 9)7 > x 5

h(x) ;

5x97v`

9x>7v`

h(x) fs bmb kmbmtmbfe. O`be` ft fs bmt fbdètfv`.Hmr h(x) tm a` surdètfv`, I somujl a` `quij tm fts ribc`. Ay cripo ribc` fs W1, 8[

I W1, 8[


(9>) Hmr ìeo mh to` hmjjmwfbc hubetfmbs hfbl wo`to`r ft fs mb`-mb` mr kiby-mb` ibl ijsm fbtm mr mbtm

(f) h(x) ; 1 tib x7 (/1, 5/1) Q (ff) h(x) ; 1x9

9

7 (, >) Q

(fff) h(x) ; x1 + b x

Ibsw`rs 4 (f) mb`-mb` mbtm (ff) mb`-mb` fbtm (fff) mb`-mb` mbtm

Mll ibl @v`b Hubetfmbs 4(f) Fh h (x) ; h (x) hmr ijj x fb to` lmkifb mh h, to`b h fs sifl tm a` ib `v`b hubetfmb.

`.c. h (x) ; ems x7 c (x) ; x + 5.

(ff) Fh h (x) ;h (x) hmr ijj x fb to` lmkifb mh h, to`b h fs sifl tm a` ib mll hubetfmb.`.c. h (x) ; sfb x7 c (x) ; x5 + x.

Bmt` 4 (f) I hubetfmb kiy b`fto`r a` mll bmr v`b. (`.c. h(x) ; `x , ems9x)

(ff) Fh ib mll hubetfmb fs l`hfb`l it x ; >, to`b h(>) ; >

^rmp`rt f`s mh @v`b/Mll Hubetfmb(i) \o` cripo mh `v`ry `v`b hubetfmb fs sykk`trfe iamut to` yixfs ibl toit mh `v`ry mll hubetfmb

fs sykk`trfe iamut to` mrfcfb.

Hmr `xikpj` cripo mh y ; x1 fs sykk`trfe iamut y-ixfs, wofj` cripo mh y ; x 5 fs sykk`trfe

iamut mrfcfb

(a) Ijj hubetfmbs (woms` lmkifb fs sykk trfeij iamut mrfcfb) eib a` `xpr`ss`l is to` suk mh ib

`v`b ibl ib mll hubetfmb, is hmjjmws


15/19

KI\OU


h(x) ;

(e) \o` mbjy hubetfmb wofeo fs l`hfb`l mb to` `btfr` buka`r jfb` ibl fs `v`b ibl mll it to` sik`

tfk` fs h(x) ; >.

(l) Fh h ibl c amto ir `v`b mr amto ir` mll, to`b to` hubetfmb h.c wfjj a` `v`b aut fh iby mb` mh to`k

fs mll ibl to` mto`r `v`b to`b h.c wfjj a` mll.

(`) Fh h(x) fs v`b to`b h(x) fs mll wofj` l`rfvitfv` mh mll hubetfmb fs `v`b. Bmt` toit sik` eibbmt a`sifl hmr fbt`crij mh hubetfmbs.

@xikpj` # 9< 4Uomw toit jmc

9xx 1 fs ib mll hubetfmb.

Umjutfmb 4 J`t h(x) ; jmc

9xx 1 .

\o`b h(x) ; jmc

9)x(x 1

; jmc

x9x

x9xx9x

1

11

; jmcx9x

9

1 ; jmc

9xx 1 ; h(x)

mr h(x) + h(x) ; >

O`be` h(x) fs ib mll hubetfmb.

@xikpj` # 90 4Uomw toit ix +ix fs ib `v`b hubetfmb.

Umjutfmb 4 J`t h(x) ; ix + ix

\o`b h(x) ; ix + i(x) ; ix +ix ; h(x).O`be` h(x) fs ib `v`b hubetfmb

@xikpj` # 98 4Uomw toit ems9 x fs b`fto`r mll bmr `v`b.

Umjutfmb 4 J`t h(x) ; ems9x. \o`b h(x) ; ems9 (x) ; ems9 x wofeo fs b`fto`r `quij tm h(x) bmr `quijtm h(x).

O`be` ems9 x fs b`fto`r mll bmr `v`b


(99) L`t`rkfb` wo`to`r to` hmjjmwfbc hubetfmbs ir` `v`b mr mll=

(f) xx

xx

````

(ff) jmc

x9x1

(fff) x jmc

9xx 1 (fv) sfb9 1x 1x9

Ibsw`rs (f) Mll (ff) Mll

(fff) @v`b (fv) Mll

@v`b `xt`bsfmb / Mll `xt`bsfmb 4J`t h(x) a` l`hfb`l fb Wi, a[ wo`r` ia>. @v`b `xt`bsfmb mh tofs hubetfmb fkpjf`s tm l`hfb` to` hubetfmbfb Wa, i[ tm kig` ft `v`b. Fb mrl`r tm c`t `v`b `xt`bsfmb r`pjie` x ay x fb to` cfv`b l`hfbftfmb.

Ufkfjirjy, mll `xt`bsfmb fkpjf`s tm l`hfb` to` hubetfmb fb Wa, i[ tm kig` ft mll. Fb mrl`r tm c`t mll`xt`bsfmb, kujtfpjy to` l`hfbftfmb mh `v`b `xt`bsfmb ay 9


16/19

KI\OU


@xikpj` # 93 4Soit fs `v`b ibl mll `xt`bsfmbs mh h(x) ; x5 8x1 + 0x 99, x >Umjutfmb 4 @v`b `xt`bsfmb mh h(x) 4

h(x) ; x5 8x1 0x 99 7 x : >

Mll `xt`bsfmb mh h(x) 4

h(x) ; x5 + 8x1 + 0x + 99 7 x : >

^`r fmlfe Hubetfmbs 4I hubetfmb h(x) fs eijj`l p`rfmlfe wfto i p`rfml \ f h to`r` `xfsts i rìj buka`r \ 2 > sueo toit hmr ìeo x

fb to` lmkifb mh h to` buka`rs x \ ibl x + \ ir ijsm fb to` lmkifb mh h ibl h(x) ; h(x + \) hmr ijj x fb

to` lmkifb mh h(x). Cripo mh i p`rfmlfe hubetfmb wfto p`rfml \ fs r`pìt`l iht`r `v`ry fbt`rvij mh '\'.

`.c. \o` hubetfmb sfb x ibl ems x amto ir` p`rfmlfe mv`r 1 ibl tib x fs p`rfmlfe mv`r\o` jìst pmsftfv` p`rfml fs eijj`l to` prfbefpij mr hublik`btij p`rfml mh h(x) mr sfkpjy to`

p`rfml mh to` hubetfmb.

Bmt` 4 Fbv`rs` mh i p`rfmlfe hubetfmb lm`s bmt `xfst.

^ rm p` rt f` s mh ^`r fmlfe Hu betfmb s 4(i) Fh h(x) ois i p`rfml \, to`b )x(h

9ibl )x(h ijsm oiv` i p`rfml \.\.

(a) Fh h(x) ois i p`rfml \, to`b h (ix + a) ois i p`rfml|i|

\.

(e) @v`ry embstibt hubetfmb l`hfb`l hmr ijj rìj x, fs ijwiys p`rfmlfe, wfto bm hublik`btij p`rfml.

(l) Fh h (x) ois i p`rfml \9ibl c (x) ijsm ois i p`rfml \1to`b p`rfml mh h(x) c(x) mr h(x) . c(x) mr

)x(c

)x(hfs J.E.K. mh \9ibl \1prmvfl`l to`fr J.E.K. `xfsts. Omw`v`r toit J.E.K. (fh `xfsts) b``l

bmt tm a` hublik`btij p`rfml. Fh J.E.K. lm`s bmt `xfsts to`b h(x) c(x) mr h(x) . c(x) mr

)x(c

)x(hfs

bmbp`rfmlfe.

J.E.K. mh

k

,q

p,

a

i ;

k)q,( a,O.E.H.

)p,J.E.K.( i,

`.c. |sfbx| ois to` p`rfml, | emsx | ijsm ois to` p`rfml

|sfbx| + |emsx| ijsm ois i p`rfml. Aut to` hublik`btij p`rfml mh |sfbx| + |emsx| fs1

.

( ) Fh c fs i hubetfmb sueo toit cmh fs l`hfb`l mb to` lmkifb mh h ibl h fs p`rfmlfe wfto \, to`b cmh fs

ijsm p`rfmlfe wfto \ is mb` mh fts p`rfmls.

@xikpj` # 9? 4Hfbl p`rfml mh to` hmjjmwfbc hubetfmbs

(f) h(x) ; sfb1

x+ ems

5

x(ff) h(x) ; {x} + sfb x, wo`r` {.}l`bmt`s hrietfmbij

pirt hubetfmb

(fff) h(x) ; ems x . ems 5x (fv) h(x) ; sfb1

x5 ems

5

x tib

5

x1

Umjutfmb 4 (f) ^`rfml mh sfb1

xfs


17/19

KI\OU


ft fs ip`rfmlfe(fff) h(x) ; ems x . ems 5x

p`rfml mh h(x) fs J.E.K. mh

5

1,1 ; 1

aut 1kiy mr kiy bmt a` hublik`btij p`rfmlfe, aut hublik`btij p`rfml ;b

1, wo`r`

b B. O`be` ermss-eo`egfbc hmr b ; 9, 1, 5, ....w` hfbl tm a` hublik`btij p`rfmlh( + x) ; ( ems x) ( ems 5x) ; h(x)

(fv) ^`rfml mh h(x) fs J.E.K. mh1/5

1,

5/9

1,

5/1

; J.E.K. mh

5


18/19

KI\OU

"kibfsogukirpoysfes.fb" 9?

@xikpj` # 96 4(f) L t`rkfb` wo`to`r h(x) ; ; h(9)

h fs bmt mb` mb` h fs bmt fbv`rtfaj`

(fff)

h(x) ; h9(x) fs `qufvij`bt tm h(x) ; x

x1 + 1x ; x x(x + 9) ; > x ; >, 9O`be` twm smjutfmb hmr h(x) ; h9(x)

(fv) y ; 9 x1 5x + 9 ; 9 x (x 5) ; > x ; >, 5

Aut x 1 x ; 5Bmw c(h(x)) ; x

Lfhh`r`btfitfbc amto sfl`s w.r.t. x

c(h(x)). h(x) ; 9 c(h(x)) ;)x(h

9

c(h(5)) ;)5(h

9

c(9) ; ; 589

;

5

9(Is h(x) ; 1x 5)

I j t ` r b i t ` K ` t o m l

y ; x1 5x + 9

x1 5x +9 y ; >

x ;1

)y9(


19/19

KI\OU

"kibfsogukirpoysfes fb" 96

x ;1

y} ibl h(1) ; 6,

to`b hfbl h (5)

Umjutfmb 4 h(x) ; 9 xb

Is h(1) ; 6 h(x) ; 9 + x5

O`be` h(5) ; 9 + 55 ; 1?


(9} ibl

h(5) ; ?, to`b hfbl h() >, to`b prmv` toit to` hubetfmb, c(x) ;)x(h9

)x(h1

fs

ib `v`b hubetfmb.

Ibsw`r 4 (9

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Function Theory E