S`ûHBKBMB`2b- /û}MBiBQM /2 H i`Mb7Q`KiBQM LX

/2pQB` bm`p2BHHö n e U9?V

1t2`+B+2

PM +QMbB/ĕ`2 M<sup>2</sup>(R)HǶ2bT+2 p2+iQ`B2H 2m+HB/B2M /2b Ki`B+2b +``û2b /ǶQ`/`2 k ¨ +Q2{+B2Mib `û2Hb KmMB /m T`Q/mBi b+HB`2 +MQMB[m2 /û}MB TQm`A2iB Ki`B+2b /2M²(R)T` ,

(A|B) = trace(^tA·B).

RX aBA= Å a b

c d

ã2iA⁰=

Å a⁰ b⁰ c⁰ d⁰

ãbQMi /2mt Ki`B+2b /2M<sup>2</sup>(R)- [m2 pmi H2 `û2H(A|A⁰)\

kX PM MQi2T H2 bQmb@2bT+2 p2+iQ`B2H 7Q`Kû /2b Ki`B+2b i`BM;mHB`2b bmTû`B2m`2b /2M<sup>2</sup>(R)X .QMM2`- TQm` H2 T`Q/mBi b+HB`2 +MQMB[m2- mM2 #b2 Q`i?QMQ`Kû2 /2T 2i /2 bQM Q`i?Q;QMHT^?X jX aB A=

Å 1 2 3 4

ã

- /ûi2`KBM2` H2 T`QD2iû Q`i?Q;QMH /2 H Ki`B+2b Abm` T- BMbB [m2 H /BbiM+2 /2 H Ki`B+2A ¨TX

S`Q#HĕK2

.Mb iQmi +2 T`Q#HĕK2- QM MQi2 ,

@F(R⁺,R)HǶ2Mb2K#H2 /2b TTHB+iBQMb /2R⁺ /MbRc

@ E HǶ2Mb2K#H2 /2b 7QM+iBQMb f : R⁺ ! R- +QMiBMm2b- i2HH2b [m2- TQm` iQmi x > 0 `û2H- H 7QM+iBQM t7 !f(t)e ^xt bQBi BMiû;`#H2 bm`R⁺c

@F HǶ2Mb2K#H2 /2b 7QM+iBQMb +QMiBMm2b 2i #Q`Mû2b bm`R⁺X

SQm` iQmi f /Mb E- QM TT2HH2 i`Mb7Q`Kû2 /2 GTH+2 /2 f 2i QM MQi2 L(f) H 7QM+iBQM /û}MB2 TQm` iQmix >0 `û2H T` ,

L(f)(x) = Z +1

0

f(t)e ^xtdt.

RX Zm2biBQM T`ûHBKBMB`2

aQB2Mia2R2if : [a,+1[ !RmM2 7QM+iBQM +QMiBMm2 T` KQ`+2mtX SQm` iQmix/Mb[a,+1[- QM TQb2 ,F(x) =

Z x a

f(t)dtX PM +QMbB/ĕ`2 H2b T`QTQbBiBQMb bmBpMi2b ,

UBVf 2bi BMiû;`#H2 bm`[a,+1[c UBBVF /K2i mM2 HBKBi2 }MB2 2M+1X

.QMM2`- bMb /ûKQMbi`iBQM- iQmi2b H2b BKTHB+iBQMb TQbbB#H2b 2Mi`2 UBV 2i UBBV HQ`b[m2 , UV f 2bi TQbBiBp2 bm`[a,+1[c

U#V f MǶ2bi Tb TQbBiBp2 bm`[a,+1[X

R

S`iB2 A , 1t2KTH2b 2i T`QT`Bûiûb kXUV .ûKQMi`2` [m2E 2bi mM bQmb@2bT+2 p2+iQ`B2H /2F(R⁺,R)X

U#V .ûKQMi`2` [m2F 2bi mM bQmb@2bT+2 p2+iQ`B2H /2EX

U+V CmbiB}2` [m2L 2bi mM2 TTHB+iBQM HBMûB`2 /2E /MbF(R^+⇤,R)- 2bT+2 p2+iQ`B2H /2b TTHB+@

iBQMb /2]0,+1[/MbRX

jXUV PM +QMbB/ĕ`2U :R<sup>+</sup> !R/û}MB2 T`U(t) = 1X .ûi2`KBM2`L(U)X

U#V aQBi >0`û2HX PM +QMbB/ĕ`2h : [0,+1[ !R/û}MB2 TQm` iQmit>0`û2H T` , h (t) =e ^t.

.ûKQMi`2` [m2h 2bi /Mb E 2i /ûi2`KBM2`L(h )X

9X aQB2Mif /MbE 2in/MbNX PM +QMbB/ĕ`2gn:t7 !t<sup>n</sup>f(t)/2[0,+1[/MbRX SQm`x >0- DmbiB}2` HǶ2tBbi2M+2 /2 A >0i2H [m2 t<sup>n</sup>e <sup>xt</sup> 6e <sup>xt2</sup> TQm` iQmit>AX 1M /û/mB`2 [m2gn 2bi mM ûHûK2Mi /2EX

8X h`Mb7Q`Kû2 /2 GTH+2 /ǶmM2 /û`Bpû2

aQBif /MbE /2 +Hbb2C¹- +`QBbbMi2 2i #Q`Mû2 bm` [0,+1[X .ûKQMi`2` [m2f<sup>0</sup> 2bi 2M+Q`2 /MbE 2i [m2 HǶQM ,

8x2]0,+1[,L(f⁰)(x) =xL(f)(x) f(0).

eX _û;mH`Biû /ǶmM2 i`Mb7Q`Kû2 /2 GTH+2

UV .ûKQMi`2` [m2- TQm` iQmi f /MbE- H 7QM+iBQM L(f)2bi /2 +Hbb2C<sup>1</sup> bm`]0,+1[2i [m2 HǶQM ,L(f)⁰= L(g1)Qɍg1 2bi /û}MB2 ¨ H [m2biBQM9X

U#V .ûKQMi`2` [m2- TQm` iQmi f /Mb E- H 7QM+iBQM L(f) 2bi /2 +Hbb2 C<sup>1</sup> bm` ]0,+1[ 2i TQm`

x >02i n2N- /ûi2`KBM2`L(f)⁽ⁿ⁾(x)¨ HǶB/2 /ǶmM2 i`Mb7Q`Kû2 /2 GTH+2X

S`iB2 AA , *QKTQ`i2K2Mib bvKTiQiB[m2b /2 H i`Mb7Q`Kû2 /2 GTH+2 .Mb iQmi2 +2ii2 T`iB2-f 2bi mM ûHûK2Mi /2E

dX PM bmTTQb2 /Mb +2ii2 [m2biBQM [m2f 2bi /MbFX UV .ûi2`KBM2` H HBKBi2 2M+1/2L(f)X

U#V h?ûQ`ĕK2 /2 H pH2m` BMBiBH2

PM bmTTQb2- /2 THmb- [m2 f 2bi /2 +Hbb2 C¹ 2i +`QBbbMi2 bm` R⁺- p2+ f⁰ #Q`Mû2 bm` R⁺X .ûKQMi`2` [m2 lim

x!+1xL(f)(x) =f(0)X 3X h?ûQ`ĕK2 /2 H pH2m` }MH2

PM bmTTQb2 /Mb +2ii2 [m2biBQM [m2 lim

t!+1f(t) =HQɍH2bi mM `û2HX aQBi(an)<sub>n2N</sub>mM2 bmBi2 /2 `û2Hb bi`B+i2K2Mi TQbBiB7b [mB +QMp2`;2 p2`b yX

UV .ûKQMi`2` [m2f TT`iB2Mi ¨FX

U#V aQBi n mM 2MiB2` Mim`2HX .ûKQMi`2` [m2 anL(f)(an) = Z +1

0

hn(x)dxQɍ hn 2bi H 7QM+iBQM /û}MB2 bm` [0,+1[ T` ,hn(x) =e ^xf

Å x an

ãX

U+V 1M /û/mB`2- ¨ HǶB/2 /m i?ûQ`ĕK2 /2 +QMp2`;2M+2 /QKBMǶ22- [m2 , lim

n!+1anL(f)(an) =HX U/V GQ`b[m2 H6= 0- /ûi2`KBM2` mM û[mBpH2Mi /2L(f)(x)2M0X

k

NX .Mb +2ii2 [m2biBQM- QM bmTTQb2 [m2f 2bi BMiû;`#H2 bm`R⁺2i QM TQb2 ,R(x) = Z +1

x

f(t)dtTQm`

iQmix/Mb[0,+1[X

UV .ûKQMi`2` [m2R 2bi mM2 7QM+iBQM /2 +Hbb2C¹ bm`[0,+1[2i /ûi2`KBM2`R<sup>0</sup>X 1M /û/mB`2 [m2- TQm` iQmi x >0 `û2H- QM ,L(f)(x) =R(0) xL(R)(x)X U#V PM }t2">0X

CmbiB}2` /2 HǶ2tBbi2M+2 /2 A`û2H TQbBiB7 i2H [m2 TQm` iQmit>A- QM Bi , |R(t)|6"X 1M /û/mB`2 [m2- TQm` iQmi x >0- QM ,

|L(f)(x) R(0)|6x Z A

0 |R(t)|dt+".

U+V .ûKQMi`2` [m2L(f)b2 T`QHQM;2 T` +QMiBMmBiû 2M0UQM T`û+Bb2` H pH2m` 2M0 /2 +2 T`QHQM@

;2K2MiVX

S`iB2 AAA , TTHB+iBQM RyX *H+mH /2 HǶBMiû;`H2 /2 .B`B+?H2i

A+B-f 2bi H 7QM+iBQM /û}MB2 T` ,f(0) = 12i f(t) =sin(t)

t TQm` t >0`û2HX UV .ûKQMi`2` [m2 H 7QM+iBQMF :R⁺ !R/û}MB2 T`F(x) =

Z x 0

f(t)dt/K2i mM2 HBKBi2 }MB2

`û2HH2H2M+1X

U#V 1M +QMbB/û`Mi H bû`B2 X

n>0

un Qɍun=

Z (n+1)⇡

n⇡ |f(t)|dt- /ûKQMi`2` [m2f MǶ2bi Tb BMiû;`#H2 bm`R⁺X

U+V aQBix >0X .ûKQMi`2`- 2M /ûiBHHMi H2b +H+mHb- [m2- TQm` iQmiX >0- QM , Z X

0

sin(t)e ^xt dt= 1

1 +x² e ^xX(xsin(X) + cos(X)) 1 . .ûKQMi`2` [m2 H 7QM+iBQMt7 !sin(t)e ^xt 2bi BMiû;`#H2 bm`R⁺X

.ûi2`KBM2` HQ`bZ +1 0

sin(t)e ^xtdtX

U/V .ûi2`KBM2`- TQm`x >0- mM2 2tT`2bbBQM bBKTH2 /2L(f)(x)2i 2M /û/mB`2HX

SQm` +2H- QM TQm`` miBHBb2` H2 `ûbmHii bmBpMi UH /ûK`+?2 /2 H T`2mp2 ûiMi B/2MiB[m2 ¨ +2HH2 /2 H [m2biBQMNV , HQ`b[m2f /MbE pû`B}2 , lim

x!+1

Z x 0

f(t)dt=H2R- HQ`b lim

x!0L(f)(x) =HX PM MQi2` [m2- T` `TTQ`i ¨ H [m2biBQMN- QM `2KTH+û HǶ?vTQi?ĕb2 dzf BMiû;`#H2 bm`R<sup>+</sup>Ǵ T` HǶ?vTQi?ĕb2 dz lim

x!+1

Z x 0

f(t)dt=H2RǴX

j

/2pQB` bm`p2BHHö n 8 U9?V

@ .Mb H2 T`Q#HĕK2- /ûbB;M2 iQmDQm`bmM2 TTHB+iBQM +QMiBMm2 /2R⁺ /MbR⁺- +`QBbbMi2 2i MQM KDQ`û2X

@ .Mb H2 T`Q#HĕK2- f /ûbB;M2iQmDQm`bmM2 TTHB+iBQM +QMiBMm2 /2R⁺ /MbRX

@ PM MQi2EHǶ2Mb2K#H2 /2b `û2HbxTQm` H2b[m2Hb HǶTTHB+iBQMt7!f(t)e ^(t)x2bi BMiû;`#H2 bm`R⁺X

@ PM MQi2E⁰ HǶ2Mb2K#H2 /2b `û2HbxTQm` H2b[m2Hb HǶBMiû;`H2 R+1

0 f(t)e ^(t)xdt+QMp2`;2X

PM b2 T`QTQb2 +B@T`ĕb /Ƕûim/B2` H i`Mb7Q`KiBQMf 7!Lf /û}MB2M 2M AX- /Ƕ2M ûi#HB` [m2H[m2b T`QT`Bû@

iûb- /Ƕ2tKBM2` +2`iBMb 2t2KTH2b 2i /ǶmiBHBb2` H i`Mb7Q`KiBQMLTQm` HǶûim/2 /ǶmM QTû`i2m`X

S`ûHBKBMB`2b- /û}MBiBQM /2 H i`Mb7Q`KiBQM LX

AXX Zm2HH2 BM+HmbBQM 2tbBbi2@i@BH 2Mi`2 H2b 2Mb2K#H2bE 2iE<sup>0</sup>\ .ûbQ`KBb- TQm`x2E<sup>0</sup>- QM MQi2`

Lf(x) = Z +1

0

f(t)e ^(t)xdt

AX"X JQMi`2` [m2 bBE MǶ2bi Tb pB/2- HQ`bE 2bi mM BMi2`pHH2 MQM KDQ`û /2RX AX*X JQMi`2` [m2 bBE MǶ2bi Tb pB/2- HQ`b Lf 2bi +QMiBMm2 bm`EX

1t2KTH2b /Mb H2 +b /2 f TQbBiBp2X

AAXX *QKT`2`E 2iE⁰ /Mb H2 +b Qɍ f 2bi TQbBiBp2X AAX"X .Mb H2b i`QBb +b bmBpMib- /ûi2`KBM2`EX AAX"XRV f(t) = ⁰(t)p2+ bmTTQbû2 /2 +Hbb2C¹X AAX"XkV f(t) =e^{t (t)}X

AAX"XjV f(t) =^e_1+t^{t (t)}2 X

AAX*X .Mb +2ii2 [m2biBQM- QM ûim/B2 H2 +b (t) =t² 2if(t) = _1+t¹2 TQm` iQmit2R<sup>+</sup>X AAX*XRV .ûi2`KBM2`EX Zm2 pmi Lf(0)\

AAX*XkV S`Qmp2` [m2Lf 2bi /û`Bp#H2 bm`R^+⇤X

AAX*XjV JQMi`2` HǶ2tBbi2M+2 /ǶmM2 +QMbiMi2A >0 i2HH2 [m2 TQm` iQmix >0- QM Bi

Lf(x) (Lf)⁰(x) = A px AAX*X9V PM MQi2g(x) =e ^xLf(x)TQm`x 0X JQMi`2` [m2

8x 0, g(x) =⇡

2 A

Z x 0

e ^t pt dt AAX*X8V 1M /û/mB`2 H pH2m` /2 HǶBMiû;`H2R+1

0 e ^t2dtX 9

1im/2 /ǶmM T`2KB2` 2t2KTH2X

.Mb +2ii2 T`iB2- (t) =tTQm` iQmit2R⁺ 2if(t) =_et^t1 1 + ^t₂ TQm` iQmit2R^+⇤X

AAAXX JQMi`2` [m2f b2 T`QHQM;2 T` +QMiBMmBiû 2M0X PM MQi2 2M+Q`2f H2 T`QHQM;2K2Mi Q#i2MmX AAAX"X .ûi2`KBM2`EX

AAAX*X HǶB/2 /ǶmM /ûp2HQTT2K2Mi 2M bû`B2- KQMi`2` [m2 TQm` iQmix >0- QM

Lf(x) = 1 2x²

1 x+

+1X

n=1

1 (n+x)² AAAX.X 1bi@+2 [m2Lf(x) _2x¹2 +¹_x /K2i mM2 HBKBi2 }MB2 2M0⁺\

:ûMû`HBiûb /Mb H2 +b ivTB[m2X

.Mb +2ii2 T`iB2- (t) =tTQm`t2R⁺X

AoXX JQMi`2` [m2 bB E MǶ2bi Tb pB/2 2i bB↵2bi b #Q`M2 BM7û`B2m`2 UQM +QMpB2Mi [m2↵= 1bB E=RV HQ`bLf 2bi /2 +Hbb2 C¹bm` ]↵,+1[2i 2tT`BK2` b2b /û`Bpû2b bm++2bbBp2b ¨ HǶB/2 /ǶmM2 BMiû;`H2X

AoX"X .Mb H2 +b T`iB+mHB2` Qɍ f(t) =e ^attⁿ TQm` iQmit2R<sup>+</sup>- p2+n2N2ia2R- 2tTHB+Bi2`E- E⁰ 2i +H+mH2`Lf(x)TQm` x2E⁰X

AoX*X *QKTQ`i2K2Mi 2M HǶBM}MBX

PM bmTTQb2 B+B [m2 E MǶ2bi Tb pB/2 2i [m2f /K2i m pQBbBM;2 /2 0 H2 /ûp2HQTT2K2Mi HBKBiû /ǶQ`/`2n2NbmBpMi ,

f(t) = Xn

k=0

ak

k!t^k+O(tⁿ⁺¹)

AoX*XRV JQMi`2` [m2 TQm` iQmi >0- QM - HQ`b[m2xi2M/ p2`b+1- H2 /ûp2HQTT2K2Mi bvKTiQiB[m2 bmBpMi ,

Z

0

f(t) Xn

k=0

ak

k!t^k

!

e ^txdt=O(x ^{n 2})

AoX*XkV 1M /û/mB`2 [m2 HQ`b[m2xi2M/ p2`b+1- QM H2 /ûp2HQTT2K2Mi bvKTiQiB[m2 ,

Lf(x) = Xn

k=0

ak

x^k+1 +O(x ^{n 2}) AoX.X *QKTQ`i2K2Mi 2M 0X

PM bmTTQb2 B+B [m2f /K2i mM2 HBKBi2 }MB2`2M+1X AoX.XRV JQMi`2` [m2E +QMiB2Mi R^+⇤X

AoX.XkV JQMi`2` [m2xLf(x)i2M/ p2`b`2M0⁺X

1im/2 /ǶmM /2mtBĕK2 2t2KTH2X

.Mb +2ii2 T`iB2- (t) = t TQm` iQmi t 2 R⁺ 2i f(t) = ^sin(t)_t TQm` iQmi t > 0- f ûiMi T`QHQM;û2 T`

+QMiBMmBiû 2M0X

oXX JQMi`2` [m2E M2 +QMiB2Mi Tb0X oX"X JQMi`2` [m2E=]0,+1[X

oX*X JQMi`2` [m2E⁰ +QMiB2Mi0X oX.X *H+mH2`(Lf)<sup>0</sup>(x)TQm`x2EX oX1X 1M /û/mB`2(Lf)(x)TQm` x2EX

8

oX6X PM MQi2 TQm` n2N2i x 0- fn(x) =R(n+1)⇡

n⇡

sin(t)

t e ^tx dtX JQMi`2` [m2 P(fn)n 0 +QMp2`;2 mMB7Q`KûK2Mi bm`[0,+1[X

oX:X Zm2 pmiLf(0)\

AMD2+iBpBiû /Mb H2 +b ivTB[m2X

.Mb +2ii2 T`iB2- (t) =tTQm` iQmit2R⁺X

oAXX aQBigmM2 TTHB+iBQM +QMiBMm2 /2[0,1]/MbRX PM bmTTQb2 [m2 TQm` iQmin2N- QM Z 1

0

tⁿg(t)dt= 0 oAXXRV Zm2 /B`2 /2 R1

0 P(t)g(t)dtTQm` P 2R[X]\ oAXXkV 1M /û/mB`2 [m2 g2bi HǶTTHB+iBQM MmHH2X

oAX"X aQB2Mif }tû2 i2HH2 [m2E bQBi MQM pB/2-x2E 2ia >0X PM TQb2h(t) =Rt

0e ^xuf(u)duTQm`

iQmit 0X

oAX"XRV JQMi`2` [m2Lf(x+a) =aR+1

0 e ^ath(t)dtX

oAX"XkV PM bmTTQb2 [m2 TQm` iQmi n 2 N- QM Lf(x+na) = 0X JQMi`2` [m2- TQm` iQmi n2 N- HǶBMiû;`H2R1

0 uⁿhÄ _ln(u)

a

ä du+QMp2`;2 2i [mǶ2HH2 2bi MmHH2X oAX"XjV ZmǶ2M /û/mBi@QM TQm` H 7QM+iBQMh\

oAX*X JQMi`2` [m2 HǶTTHB+iBQM [mB ¨ f bbQ+B2Lf 2bi BMD2+iBp2X

1im/2 2M H #Q`M2 BM7û`B2m`2 /2 EX

oAAXX *b TQbBiB7X

PM bmTTQb2 [m2f 2bi TQbBiBp2 2i [m2EMǶ2bi MB pB/2 MB û;H ¨RX PM MQi2↵b #Q`M2 BM7û`B2m`2X oAAXXRV JQMi`2` [m2 bBLf 2bi #Q`Mû2 bm`E- HQ`b↵2EX

oAAXXkV aB ↵2/E- [m2 /B`2 /2Lf(x)[mM/ xi2M/ p2`b↵⁺\ oAAX"X .Mb +2ii2 [m2biBQM-f(t) = cos(t)2i (t) = ln(1 +t)X oAAX"XRV .ûi2`KBM2`EX

oAAX"XkV .ûi2`KBM2`E⁰X

oAAX"XjV JQMi`2` [m2Lf /K2i mM2 HBKBi2 2M↵- #Q`M2 BM7û`B2m`2 /2E- 2i H /ûi2`KBM2`X

lM2 miBHBbiBQM /2 H i`Mb7Q`KiBQM LX

.Mb +2ii2 T`iB2- P /ûbB;M2 HǶ2Mb2K#H2 /2b 7QM+iBQMb TQHvMQKBH2b ¨ +Q2{+B2Mib `û2Hb 2i QM miBHBb2 H i`Mb7Q`KiBQMLTTHB[mû ¨ /2b ûHûK2Mib /2P TQm` HǶûim/2 /ǶmM QTû`i2m`UX

oAAAXX aQB2MiP, Q /2mt ûHûK2Mib /2PX JQMi`2` [m2 HǶBMiû;`H2R+1

0 P(t)Q(t)e ^tdt +QMp2`;2X oAAAX"X SQm` iQmi +QmTH2(P, Q)2P²- QM MQi2

(P, Q) = Z +1

0

P(t)Q(t)e ^tdt oû`B}2` [m2(., .)/û}MBi mM T`Q/mBi b+HB`2 bm`PX

oAAAX*X PM MQi2D HǶ2M/QKQ`T?BbK2 /2 /û`BpiBQM 2iU HǶ2M/QKQ`T?BbK2 /2 P /û}MB T`

U(P)(t) =e^tD(te ^tP⁰(t)) oû`B}2` [m2U 2bi 2M/QKQ`T?BbK2 /2 PX

e

oAAAX.X JQMi`2` [m2 TQm` iQmbP, Q/2P QM

(U(P), Q) = (P, U(Q))

oAAAX1X JQMi`2` [m2U /K2i /2b pH2m`b T`QT`2b /MbC- [mǶ2HH2b bQMi `û2HH2b 2i [m2 /2mt p2+i2m`b T`QT`2b bbQ+Bûb ¨ /2b pH2m`b T`QT`2b /BbiBM+i2b bQMi Q`i?Q;QMmtX

oAAAX6X aQB2Mi mM2 pH2m` T`QT`2 /2U 2iP mM p2+i2m` T`QT`2 bbQ+BûX

oAAAX6XRV JQMi`2` [m2P 2bi bQHmiBQM /ǶmM2 û[miBQM /Bzû`2MiB2HH2 HBMûB`2 bBKTH2 [m2 HǶQM T`û+Bb2`X oAAAX6XkV Zm2H HB2M v@@i@BH 2Mi`2 2i H2 /2;`û /2P\

oAAAX:X .2b+`BTiBQM /2b ûHûK2Mib T`QT`2b /2 UX PM +QMbB/ĕ`2 bm`[0,+1[ HǶû[miBQM /Bzû`2MiB2HH2

(En) : tP⁰⁰+ (1 t)P⁰+nP = 0 p2+n2N2i /ǶBM+QMMm2P 2PX

oAAAX:XRV 1M TTHB[mMi H i`Mb7Q`KiBQMLp2+ (t) =t¨(En)- KQMi`2` [m2 bBP 2bi bQHmiBQM /2 (En)bm`[0,+1[- HQ`b bQM BK;2QT`L2bi bQHmiBQM /ǶmM2 û[miBQM /Bzû`2MiB2HH2 (E⁰_n)/ǶQ`/`2 1bm`]1,+1[X

oAAAX:XkV _ûbQm/`2 HǶû[miBQM (E<sub>n</sub><sup>0</sup>) bm` ]1,+1[ 2i 2M /û/mB`2 H2b pH2m`b 2i p2+i2m`b T`QT`2b /2 HǶ2M/QKQ`T?BbK2UX

oAAAX:XjV Zm2H 2bi H2 HB2M 2Mi`2 +2 [mB T`û+ĕ/2 2i H2b 7QM+iBQMb TQHvMQKBH2b /û}MB2b TQm`n2NT`

Pn(t) =e^tDⁿ(e ^ttⁿ)\

d

*2Mi`H2 SaA R lM +Q``B;û

S`ûHBKBMB`2b- /û}MBiBQM /2 H i`Mb7Q`KiBQM LX

AXX GǶBMiû;`#BHBiû 2Mi`ŗM2 H +QMp2`;2M+2 /2 HǶBMiû;`H2 2i QM /QM+

E⇢E⁰

AX"X aB x2 E HQ`b 8y x, 8t 2 R<sup>+</sup>, |f(t)e <sup>(t)y)</sup> |f(t)|e <sup>(t)x</sup> U+` (t) 0V 2i- H2 KDQ`Mi ûiMi BMiû;`#H2 bm`R<sup>+</sup>- QM y2EX PM pB2Mi /QM+ /2 pQB` [m2

8x2E, [x,+1[⇢E PM bmTTQb2 /ûbQ`KBbE MQM pB/2 2i QM /BbiBM;m2 /2mt +bX

@ aBEMǶ2bi Tb KBMQ`û c TQm` iQmi `û2Hy BH 2tBbi2x2Ei2H [m2 xy 2i +2 [mB T`û+ĕ/2 BM/B[m2 BM/B[m2 [m2y2EX PM /QM+

E =R

@ aBE 2bi KBMQ`û- ûiMi MQM pB/2 BH TQbbĕ/2 mM2 #Q`M2 BM7û`B2m`2↵2iE⇢[↵,+1[X S` BHH2m`b- bB y > ↵HQ`b U+`+iû`BbiBQM /2 H #Q`M2 BM7û`B2m`2V BH 2tBbi2 x 2 E i2H [m2 x y 2i BMbB y2EX PM T`Qmpû [m2

]↵,+1[⇢E⇢[↵,+1[ 2i E 2bi û;H ¨ HǶmM /2b BMi2`pHH2b]↵,+1[ Qm[↵,+1[X

AX*X AH bǶ;Bi /ǶmiBHBb2` H2 i?ûQ`ĕK2 /2 +QMiBMmBiû /2b BMiû;`H2b ¨ T`Kĕi`2bX

@ 8x2E, t7!f(t)e ^(t)x2bi +QMiBMm2 bm`R⁺X

@ 8t 0, x7!f(t)e ^(t)x2bi +QMiBMm2 bm` EX

@ 8[a, b]⇢E, 8x2[a, b], 8t 0, |f(t)e ^(t)x||f(t)|e ^(t)aX G2 KDQ`Mi 2bi BM/ûT2M/Mi /2 x2i 2bi BMiû;`#H2 bm`R⁺X

G2 i?ûQ`ĕK2 bǶTTHB[m2 2i /QMM2

Lf 2C⁰(E)

1t2KTH2b /Mb H2 +b /2 f TQbBiBp2X

AAXX GǶBMiû;`#BHBiû /2 g bm` I +Q``2bTQM/Mi ¨ H +QMp2`;2M+2 /2 HǶBMiû;`H2 /2 |g| bm` I- bB f 2bi TQbBiBp2 HQ`bE=E⁰X

AAX"X .Mb H2b i`QBb +b T`QTQbûb- H 7QM+iBQM f 2bi TQbBiBp2 U2M "XRV +2H /û+QmH2 /2 H +`QBbbM+2 bmTTQbû2 /2 VX PM T2mi /QM+ BM/Bzû`2KK2Mi ûim/B2` H +QMp2`;2M+2 /2 HǶBMiû;`H2 UBX2X HǶ2tBbi2M+2 /ǶmM2 HBKBi2 /2Ra

0 f(t)e ^(t)x dt [mM/a!+1V Qm HǶBMiû;`#BHBiû Um pQBbBM;2 /2+1+` H2b 7QM+iBQMb bQMi +QMiBMm2b bm`R<sup>+</sup> 2i+12bi /QM+ H2 b2mH2 T`Q#HĕK2VX

AAX"XRV aQBia2R⁺X PM

8x6= 0, Z a

0

0(t)e ^(t)xdt= ï 1

xe ^(t)x òt=a

t=0

=e ^(0)x e ^(a)x x ûiMi +`QBbbMi2 2i MQM KDQ`û2 i2M/ p2`b+12M+12i BMbB

8x >0, lim

a!+1

Z a 0

0(t)e ^(t)xdt= e ^(0)x x 8x <0, lim

a!+1

Z a 0

0(t)e ^(t)xdt= +1

3

1M}M U+bx= 0VRa

0 0(t)dt= (a) (0)!+1[mM/ a!+1X PM /QM+ KQMi`û [m2 E=E⁰ =R^+⇤ 2i 8x >0, Lf(x) = e ^(0)x

x

AAX"XkV aQBix2RX PM 8t max(x,0), f(t)e ^(t)x=e ^{(t)(t x)} 1X PM MǶ /QM+ Tb BMiû;`#BHBiû m pQBbBM;2 /2+12i

E=;

AAX"XjV aQBix2RX PM 8t max( x,0), 0f(t)e ^(t)x= ^e _1+t^(t)(x+t)2  1+t¹²X G2 KDQ`Mi ûiMi BMiû;`#H2- QM x2EX BMbB

E=E⁰=R AAX*X AH bǶ;Bi /Ƕûim/B2` H 7QM+iBQMx7!R+1

0

e ^xt2 1+t² dtX

AAX*XRV aBx 0HQ`b0 <sup>e</sup>1+t<sup>xt22</sup>  1+t<sup>12</sup> [mB 2bi BMiû;`#H2 bm`R⁺X PM /QM+x2EX

aBx <0- t^e_1+t^xt2² !+1[mM/ t!+1U+`QBbbM+2b +QKT`û2bV 2it7! ^e1+t^xt22 MǶ2bi Tb BMiû;`#H2 m pQBbBM;2 /2+1U+QKT`BbQM ¨ H 7QM+iBQM /2 _B2KMMt7!1/tV 2i x /2EX PM /QM+

E=R⁺

PM BKKû/Bi2K2Mi UarctanûiMi mM2 T`BKBiBp2 /2t7! 1+t<sup>12</sup> bm`RV

Lf(0) = Z +1

0

dt 1 +t² =⇡

2

AAX*XkV AH bǶ;Bi /ǶmiBHBb2` H2 i?ûQ`ĕK2 /2 `û;mH`Biû /2b BMiû;`H2b ¨ T`Kĕi`2bX

@ 8x >0, 8t 0, t7! ^{e xt}

2

1+t² 2bi BMiû;`#H2 bm`R⁺X

@ 8t 0-x7! ^e1+t^xt22 2bi /2 +Hbb2C¹ bm`R<sup>+⇤</sup>/2 /û`Bpû2x7! ^t21+t^{e xt22}X

@ 8x >0, t7! ^t21+t^{e xt22} 2bi +QMiBMm2 bm`R⁺X

@ PM

8a >0, 8x a, 8t 0, t²e ^xt2

1 +t²  t²

1 +t²e ^at2 = (t)

2bi +QMiBMm2 bm`R<sup>+</sup> 2i Mû;HB;2#H2 /2pMi1/t<sup>2</sup> m pQBbBM;2 /2+1U+`a >0)c +Ƕ2bi /QM+

mM2 7QM+iBQM BMiû;`#H2 bm` R⁺X

G2 +Qm`b BM/B[m2 HQ`b [m2Lf 2bi /2 +Hbb2C¹ bm`R^+⇤ p2+

8x >0, (Lf)⁰(x) =

Z +1 0

t²

1 +t²e ^xt2 dt

_2K`[m2 , QM M2 bBi `B2M [mMi ¨ H /û`Bp#BHBiû 2M0 2i QM M2 bǶ2M Q++mT2` [m2 bB +2H bǶpĕ`2 miBH2 /Mb H bmBi2X

AAX*XjV PM 2M /û/mBi [m2

8x >0, Lf(x) (Lf)⁰(x) = Z +1

0

e^xt2 dt= A

px p2+ A= Z +1

0

e ^t2 dt H /2`MBĕ`2 û;HBiû T`Qp2MMi /m +?M;2K2Mi /2 p`B#H2u=tpxX

_2K`[m2 , Lf ûiMi +QMiBMm2 2M 0- QM /QM+ (Lf)<sup>0</sup>(x)! 1 [mM/x!0<sup>+</sup>X S` i?ûQ`ĕK2 /2 HBKBi2 /2 H /û`Bpû2 Up2+Lf 2C⁰(R⁺)\C¹(R^+⇤)- QM T2mi 2M /û/mB`2 [m2Lf MǶ2bi Sa /û`Bp#H2 2M0 KBb [m2 bQM ;`T?2 T`ûb2Mi2 2M (0,⇡/2) mM2 /2KB@iM;2Mi2 p2`iB+H2X

N

AAX*X9V g2bi /û`Bp#H2 bm`R^+⇤ 2i

8x >0, g⁰(x) =e ^x((Lf)⁰(x) Lf(x)) = Ae ^x px

x7! ^ep_x^x 2bi +QMiBMm2 bm`R<sup>+⇤</sup> 2i H2 i?ûQ`ĕK2 7QM/K2MiH BM/B[m2 [m2 x7!Rx 1

ep^t

t dt2M 2bi mM2 T`BKBiBp2 bm`RX .2mt T`BKBiBp2b bm` mM BMi2`pHH2 /Bzû`2Mi /ǶmM2 +QMbiMi2-

9c2R/8x >0, g(x) =c A Z x

1

e ^t pt dt

S` BHH2m`b-g 2bi +QMiBMm2 2M 0 ULf HǶ2biV 2ig(x)!g(0) = ^⇡₂ [mM/ x!0X PM 2M /û/mBi [m2 HǶQM T2mi Tbb2` ¨ H HBKBi2 /Mb HǶû;HBiû +B@/2bbmb U2M T`iB+mHB2`- HǶBMiû;`H2 2tBbi2 bm` [0,1] +2 [mB MǶ2bi Tb bm`T`2MMi +` H 7QM+iBQM [m2 HǶQM BMiĕ;`2 2bi T`QHQM;2#H2 T` +QMiBMmBiû 2M0VX PM Q#iB2Mic=^⇡₂ +AR0

1 ep^t

t dtX 6BMH2K2Mi- QM 8x >0, g(x) =⇡

2 A

Z x 0

e ^t pt dt 2i HǶû;HBiû `2bi2 p`B2 2Mx= 0 U2HH2 b2 HBig(0) =⇡/2VX

AAX*X8V _2K`[mQMb [m2

8x 0, 0g(x)e ^x Z +1

0

dt 1 +t²_x !

!+10

1M 7BbMi i2M/`2xp2`b+1/Mb HǶB/2MiBiû /2 H [m2biBQM T`û+û/2Mi2- QM Q#iB2Mi /QM+

x!lim+1A Z x

0

e ^t pt dt= ⇡

2 S` BHH2m`b- H2 +?M;2K2Mi /2 p`B#H2u=p

t UHB+Bi2 +` t7!p

t 2bi mMC¹ /BzûQKQ`T?BbK2 /2 ]0, x[/Mb]0,px[V /QMM2

8x >0, Z x

0

e ^t pt dt= 2

Z ^px 0

e ^u2du

*2ii2 [mMiBiû i2M/ p2`b2A[mM/x!+12i }MH2K2Mi-2A<sup>2</sup>= <sup>⇡</sup><sub>2</sub> Qm 2M+Q`2 U+QKK2A 0V Z +1

0

e ^t2dt=A= p⇡

2

1im/2 /ǶmM T`2KB2` 2t2KTH2X

AAAXX *QKK2e^t 1⇠⁰t- QM f(t)!0[mM/ t!0⁺Xf 2bi /QM+ T`QHQM;2#H2 T` +QMiBMmBiû 2M TQbMi

f(0) = 0

AAAX"X aQBix2Rcg : t7!f(t)e ^xt2bi +QMiBMm2 bm`R<sup>+</sup>2ig(t)⇠ <sup>t</sup>2e <sup>xt</sup>2M+1U+`f(t)⇠⁺1t/2VX aB x > 0- g 2bi BMiû;`#H2 m pQBbBM;2 /2 +1 UMû;HB;2#H2 /2pMi 1/t<sup>2</sup>VX aB x  0- g 2bi MQM BMiû;`#H2 m pQBbBM;2 /2+1U/2 HBKBi2 BM}MB2VX BMbB

E=R^+⇤ Ry

AAAX*X S` /û}MBiBQM- QM

8x >0, Lf(x) = Z +1

0

f(t)e ^xtdt S` BHH2m`b- TQm`t >0 QM e ^t2[0,1[2i /QM+

8t >0, f(t) =te ^t

+1X

k=0

e ^kt 1 + t 2

G2b 7QM+iBQMbt7!e ^xt2it7!te ^xtûiMi BMiû;`#H2b bm`R⁺ TQm`x >0- QM T2mi /û+QmT2`Lf(x) 2M i`QBb KQ`+2mt TQm`x >0 2i Q#i2MB`

8x >0, Lf(x) = Z +1

0 +1

X

k=0

te ^(k+1+x)tdt

Z +1 0

e ^xtdt+1 2

Z +1 0

te ^xtdt

SQm` y > 0- mM2 BMiû;`iBQM T` T`iB2b /QMM2 Ra

0 te ^yt dt = î _t

ye ^ytóa

0 + ¹_yRa

0 e ^yt dt =

aye ^ya+1 e ^ay

y² X 1M 7BbMi i2M/`2 ap2`b+1- QM i`Qmp2 HQ`b 8y >0,

Z +1 0

te ^aydt= 1 y² 2i BMbB

8x >0, Lf(x) = 1 2x²

1 x+

Z +1 0

+1X

k=0

te ^(k+1+x)tdt

PM p2mi KBMi2MMi BMi2`p2`iB` bQKK2 2i BMiû;`H2 T` H2 i?ûQ`ĕK2 /ǶBMiû;`iBQM i2`K2 ¨ i2`K2X PM i`pBHH2 TQm` mMx >0 }tûX

@ SQbQMbfk : t7!te ^(k+1+x)tĘfk 2bi +QMiBMm2 TQm` iQmik 02i P(fk)+QMp2`;2 bBKTH2K2Mi bm`R<sup>+⇤</sup> /2 bQKK2t7! e<sup>tt</sup>1e <sup>xt</sup> [mB 2bi mbbB +QMiBMm2 bm`R^+⇤X

@ G2bfk bQMi BMiû;`#H2b bm`R⁺ 2iR+1

0 |fk|= _(k+1+x)¹ 2 2bi H2 i2`K2 ;ûMû`H /ǶmM2 bû`B2 +QMp2`@

;2Mi2X

G2 i?ûQ`ĕK2 bǶTTHB[m2 2i H2 +H+mH /ǶBMiû;`H2 7Bi THmb ?mi /QMM2

8x >0, Lf(x) = 1 2x²

1 x+

+1

X

k=0

1 (k+x+ 1)²

*Ƕ2bi H 7Q`KmH2 pQmHm2 UBH bm{i /2 TQb2`n=k+ 1VX AAAX.X PM pB2Mi /2 pQB` [m2

8x >0, Lf(x) 1 2x² + 1

x=

+1

X

n=1

1 (n+x)²

@ SQbQMbhn : x7!(n+x)^{1 2}X PM khnk1,R⁺n¹² [mB 2bi H2 i2`K2 ;ûMû`H /ǶmM2 bû`B2 +QMp2`;2Mi2X BMbB-P(hn)+QMp2`;2 MQ`KH2K2Mi bm`R⁺X

@ G2bhn bQMi +QMiBMm2b bm`R⁺X

G2 +Qm`b BM/B[m2 [m2 H bQKK2 /2 H bû`B2P(hn)2bi +QMiBMm2 bm`R<sup>+</sup>X 1M T`B+mHB2`-

x!0lim⁺Lf(x) =

+1X

n=1

1 n² =⇡²

6 RR

:ûMû`HBiûb /Mb H2 +b ivTB[m2X

AoXX AH bǶ;Bi /ǶmiBHBb2` H2 i?ûQ`ĕK2 /2 `û;mH`Biû /2b BMiû;`H2b ¨ T`Kĕi`2bX

@ SQm` iQmix >↵U2i /QM+x2EVt7!e <sup>xt</sup>f(t)2bi BMiû;`#H2 bm`R⁺X

@ SQm` iQmit 0-x7!e <sup>xt</sup>f(t)2bi /2 +Hbb2C<sup>1</sup>bm`]↵,+1[/2 /û`Bpû2n@BĕK2x7!( t)ⁿe ^xtf(t)X

@ SQm` iQmix >↵-t7!( t)<sup>n</sup>e <sup>xt</sup>f(t)2bi +QMiBMm2 bm`R⁺X

@ aQB2Min2N^⇤2i [a, b]⇢]↵,+1[X PM

8x2[a, b], 8t 0, |( t)ⁿe ^xtf(t)|tⁿe ^at|f(t)|= n(t)

n 2bi +QMiBMm2 bm` R<sup>+</sup> 2i T`ûb2Mi2 mM mMB[m2 T`Q#HĕK2 /ǶBMiû;`#BHBiû 2M +1X *QKK2 a >

↵= inf(E)- BH 2tBbi2c2E i2H [m2a > cU+`+iû`BbiBQM /2 H #Q`M2 BM7û`B2m`2VX PM HQ`b- m pQBbBM;2 /2 +1- n(t) =e ^ct|f(t)|tⁿe ^{(a c)t}=o(e ^ctf(t)) U+`a c > 0VX *QKK2 c 2E- t7!e <sup>ct</sup>f(t)2bi BMiû;`#H2 m pQBbBM;2 /2+1Ubm`R<sup>+</sup>VX BMbB- n 2bi BMiû;`#H2 m pQBbBM;2 /2+12HH2 mbbB 2i /QM+ 2HH2 HǶ2bi bm`R⁺X

G2 i?ûQ`ĕK2 bǶTTHB[m2X AH BM/B[m2 [m2Lf 2C¹(]↵,+1[)2i

8n2N^⇤, 8x >↵, (Lf)ⁿ(x) = ( 1)ⁿ Z +1

0

tⁿf(t)e ^xtdt AoX"X *QKK2f 2bi TQbBiBp2- QM E=E⁰X AH bǶ;Bi /2 i`Qmp2` H2b x2 Ri2H [m2R+1

0 tⁿe ^(x+a)tdt +QMp2`;2 UH2 b2mH T`Q#HĕK2 ûiMi +2HmB m pQBbBM;2 /2+1V Qm i2Hb [m2t7!tⁿe ^(x+a)t2bi BMiû;`#H2 bm`R⁺UB/2KVX aBx+a >0HQ`bt<sup>n</sup>e <sup>(x+a)t</sup>=o(1/t<sup>2</sup>)m pQBbBM;2 /2+1U+`QBbbM+2b +QKT`û2bV 2i H 7QM+iBQM 2bi BMiû;`#H2 m pQBbBM;2 /2 +1X aB x+a 0 HQ`b t.tⁿe ^(x+a)t ! +1 [mM/

t!+12i H 7QM+iBQM MǶ2bi /QM+ Tb BMiû;`#H2 m pQBbBM;2 /2+1X 6BMH2K2Mi c E=E⁰=] a,+1[

SQm` y > 0 2i n 2 N<sup>⇤</sup>- mM2 BMiû;`iBQM T` T`iB2b /QMM2 U2M QK2iiMi H2b /ûiBHb /2 +H+mHV R+1

0 tⁿe ^ytdt= ⁿ_yR+1

0 t^{n 1}e ^ytdtX PM 2M /û/mBi T` `û+m``2M+2 bBKTH2 [m2 8y >0, 8n2N,

Z +1 0

tⁿe ^ytdt= n!

yⁿ⁺¹ 1M T`iB+mHB2`- QM

8x > a, Lf(x) = n!

(x+a)ⁿ⁺¹ AoX*XRV PM }t2 > 0X SQbQMb g : t 7! f(t) Pn

k=0ak

k!t^kX PM bBi [mǶm pQBbBM;2 /2 0 QM g(t) =O(tⁿ⁺¹)X AH 2tBbi2 /QM+M 2R⁺ 2ic2]0, [ i2H [m2

8t2[0, c], |g(t)|M tⁿ⁺¹

G `2HiBQM /2 *?bH2b- HǶBMû;HBiû i`BM;mHB`2 2i H +`QBbbM+2 /m Tbb;2 ¨ HǶBMiû;`H2 /QMM2Mi Z

0

g(t)e ^txdt  Z c

0 |g(t)|e ^tx dt+kgk1,[c, ]

Z

c

e ^xt dtM Z c

0

tⁿ⁺¹e ^txdt+kgk1,[c, ]( c)e ^cx G2 +H+mH /2 AoX" BM/B[m2 HQ`b [m2

Z

0

g(t)e ^txdt M(n+ 1)!

xⁿ⁺² +kgk1,[c, ]( c)e ^cx=Ox!+1(x ^{n 2}) +2 [mB +Q``2bTQM/ m `ûbmHii /2KM/ûX

Rk

AoX*XkV p2+ H2 +H+mH /2 AoX" QM U2M +QMiBMmMi ¨ miBHBb2` H 7QM+iBQMg BMi`Q/mBi2 2M AoX*XRV

8x2E\R^+⇤, Lf(x) Xn

k=0

ak

x^k+1 = Z +1

0

g(t)e ^txdt

G [m2biBQM T`û+û/2Mi2 /QMM2 mM `2Mb2B;M2K2Mi TQm` HǶBMiû;`H2 2Mi`20 2i 1X AMiû`2bbQMb@MQmb ¨ HǶmi`2 T`iB2X 6BtQMbc 2E\R^+⇤ Uc 2tBbi2 +` E 2bi MQM KDQ`ûV- i`pBHHQMb p2+ x c+ 12i û+`BpQMb [m2

8t 1, |g(t)e ^tx|=|g(t)e ^cte ^{(x c)t}||g(t)e ^ct|e ^{(x c)}

t7!g(t)e ^ct 2bi BMiû;`#H2 bm`[1,+1[U+`c2E /QM+t7!f(t)e <sup>ct</sup>2bi BMiû;`#H2 2i c >0 /QM+

TQm` iQmik-t7!t<sup>k</sup>e <sup>ct</sup>2bi mbbB BMiû;`#H2 bm` R<sup>+</sup>VX PM HQ`b Z +1

1

g(t)e ^xtdt e ^{(x c)} Z +1

1 |g(t)|e ^ctdt

2i +2i i2`K2 2bi /QKBMû T` U2i KāK2 Mû;HB;2#H2 /2pMiVx ^{n 2} [mM/ x!+1X 1M bQKKMi- QM HQ`b

Z +1 0

f(t) Xn

k=0

ak

k!t^k

!

e ^tx dt=O(x ^{n 2}) 2i H2 +H+mH /2 AoX" /QMM2

Lf(x) = Xn

k=0

ak

x^k+1 +O(x ^{n 2})

AoX.XRV f 2bi +QMiBMm2 bm` R<sup>+</sup> 2i /K2i mM2 HBKBi2 2M+12i 2HH2 2bi /QM+ #Q`Mû2 bm`R<sup>+</sup> UKDQ`û2 2M KQ/mH2 T` |`|+ 1 m pQBbBM;2 /2 +1 2i +QMiBMm2 bm` H2 b2;K2Mi [mB `2bi2VX aQBi x > 0c

|f(t)e ^xt| kfk1,R⁺e ^xt 2bi BMiû;`#H2 bm`R⁺ 2i /QM+x2EX BMbB R^+⇤⇢E

AoX.XkV PM

8x >0, xLf(x) = Z +1

0

xf(t)e ^xtdt G2 +?M;2K2Mi /2 p`B#H2u=xt/QMM2

8x >0, xLf(x) = Z +1

0

f(u/x)e ^udu=G(x)

SQm` ûim/B2` H2 +QKTQ`i2K2Mi /2 G2M +1- QM p miBHBb2` H +`+iû`BbiBQM bû[m2MiB2HH2X PM b2 /QMM2 BMbB mM2 bmBi2(xn)/ǶûHûK2Mib /2]0,1]i2HH2 [m2xn!02i QM p2mi KQMi`2` [m2G(xn)!`X SQm` +2H- QM miBHBb2 H2 i?ûQ`ĕK2 /2 +QMp2`;2M+2 /QKBMû2X

@ SQbQMbgn : u7!f(u/xn)e ^uX(gn)2bi mM2 bmBi2 /2 7QM+iBQMb +QMiBMm2b [mB +QMp2`;2 bBKTH2@

K2Mi bm`R<sup>+⇤</sup> p2`b H 7QM+iBQM +QMbiMi2u7!`2HH2 KāK2 +QMiBMm2 bm`R⁺X

@ 8n2N, 8u 0, |gn(u)| kfk1e ^u2i H2 KDQ`Mi 2bi BMiû;`#H2 bm` R<sup>+</sup>X G2 i?ûQ`ĕK2 bǶTTHB[m2 2i BM/B[m2 [v2G(xn)!R+1

0 `e <sup>u</sup>du=`X PM Q#iB2Mi H KāK2 HBKBi2 TQm`

iQmi2b H2b bmBi2b(xn)2i BMbB U+`+iû`BbiBQM bû[m2MiB2HH2 /2b HBKBi2bV

x!0lim⁺xLf(x) =` Rj

1im/2 /ǶmM /2mtBĕK2 2t2KTH2X

oXX AH bǶ;Bi /2 KQMi`2` [m2f MǶ2bi Tb BMiû;`#H2 bm`R⁺Qm 2M+Q`2 [m2F(a) =Ra

0 |f|MǶ/K2i Tb /2 HBKBi2 BM}MB2 [mM/a!+1X PM `2K`[m2 [m2

8n2N,

Z (n+1)⇡

n⇡ |f(t)|dt

Z (n+1)⇡ ⇡/4 n⇡+⇡/4

pdt 2t

p 1

2(n+ 1)⇡

⇡ 2

PM 2M /û/mBi [m2

8n2N^⇤, F((n+ 1)⇡) 1 2p

2 Xn

k=0

1

k+ 1_n!+1! +1 F MǶ2bi /QM+ Tb #Q`Mû2 bm`R⁺ 2i

02/E

oX"X p2+ H T`iB2 T`ûHBKBMB`2- QM 2M /û/mBi [m2 E ⇢]0,+1[X _û+BT`Q[m2K2Mi- bB x > 0 HQ`b f(t)e ^xt=o(1/t²)m pQBbBM;2 /2+12i /QM+x2EX BMbB

E=]0,+1[ oX*X G2 b2mH T`Q#HĕK2 /Mb HǶBMiû;`H2R+1

0 f(t)dt2bi +2HmB m pQBbBM;2 /2+1X PM 8a 1,

Z a 1

sin(t) t dt=

ï cos(t) t

òa 1

+ Z a

1

cos(t) t² dt

G2 i2`K2 dziQmi BMiû;`ûǴ /m K2K#`2 /2 /`QBi2 /K2i mM2 HBKBi2 [mM/ a ! +1X t 7! ^cos(t)t² 2bi BMiû;`#H2 m pQBbBM;2 /2 +1 UKDQ`û2 2M KQ/mH2 T`1/t<sup>2</sup>) 2i HǶBMiû;`H2 /m K2K#`2 /2 /`QBi2 /K2i /QM+ mM2 HBKBi2 [mM/ a!+1VX AH 2M 2bi }MH2K2Mi /2 KāK2 /2 HǶBMiû;`H2 /m K2K#`2 /2 ;m+?2X GǶBMiû;`H2 /2 ^sin(t)_t 2tBbi2 /QM+ mbbB m pQBbBM;2 /2 +1X PM }MH2K2Mi 2tBbi2M+2 /2R+1

0

sin(t) t dt 2i

02E⁰

oX.X AH +QMpB2Mi 2M+Q`2 /ǶmiBHBb2` H2 i?ûQ`ĕK2 /2 `û;mH`Biû /2b BMiû;`H2b ¨ T`Kĕi`2bX

@ 8x >0, t7!f(t)e ^xt 2bi BMiû;`#H2 bm`R⁺U+`x2EVX

@ 8t >0, x7!f(t)e ^xt 2bi /2 +Hbb2C¹bm`R<sup>+⇤</sup> /û /û`Bpû2x7! sin(t)e ^xtX

@ 8x >0, t7! sin(t)e ^xt 2bi +QMiBMm2 bm`R⁺X

@ 8a >0, 8x a, 8t 0, | sin(t)e ^xt|e ^at [mB 2bi BMiû;`#H2 bm`R⁺X BMbB-Lf 2C¹(R^+⇤)2i

8x >0 (Lf)⁰(x) =

Z +1 0

sin(t)e ^xt dt= 1 1 +x²

H2 +H+mH /2 HǶBMiû;`H2 b2 7BbMi- T` 2t2KTH2- 2M BMi2`T`ûiMi H2 bBMmb +QKK2 T`iB2 BK;BMB`2 /2 e^itX

oX1X .2mt T`BKBiBp2b /ǶmM2 7QM+iBQM bm` mM BMi2`pHH2 /Bzû`Mi /ǶmM2 +QMbiMi2- 9c2R/8x >0, Lf(x) =c arctan(x)

f ûiMi #Q`Mû2 bm`R⁺ U+QMiBMm2 2i /2 HBKBi2 MmHH2 2M HǶBM}MBV QM |Lf(x)| kfk1R+1

0 e ^xtdt=

|fk1

x !0 [mM/x!+1X PM 2M /û/mBi [m2 c=⇡/22i 8x >0, f(x) =⇡

2 arctan(x) R9

oX6X aQBix 0 U2i /QM+x2E⁰VX G2 +?M;2K2Mi /2 p`B#H2u=x n⇡/QMM2 fn(x) = ( 1)ⁿe ^n⇡x

Z ⇡ 0

sin(u)

u+n⇡e ^uxdx

PM `2K`[m2 [m2 (fn(x)) 2bi mM2 bmBi2 Hi2`Mû2- /2 HBKBi2 MmHH2 U+` x 2 E⁰V 2i [m2 (|fn(x)|) /û+`Qŗi U+Ƕ2bi H2 +b /2b bmBi2b TQbBiBp2b /2 i2`K2 ;ûMû`mte ^n⇡x 2i R⇡

0 sin(u)

u+n⇡e ^ux dxVX PM T2mi HQ`b TTHB[m2` H `ĕ;H2 bTû+BH2 TQm` {`K2` [m2

8n2N, X

k n+1

fk(x) |fn+1(x)|

Z (n+2)⇡

(n+1)⇡

dt t = ln

Ån+ 2 n+ 1

ã

2i QM /QM+

X

k n+1

fk 1,R⁺

ln Ån+ 2

n+ 1 ã

!0 +2 [mB KQMi`2 [m2P(fk)+QMp2`;2 mMB7Q`KûK2Mi bm`R⁺X

oX:X PM T2mi BMbB miBHBb2` H2 i?ûQ`ĕK2 /2 /Qm#H2 HBKBi2 TQm` {`K2` [m2

⇡ 2 = lim

x!0Lf(x) = lim

x!0 +1

X

n=0

fn(x) =

+1

X

n=0

fn(0) =Lf(0)

AMD2+iBpBiû /Mb H2 +b ivTB[m2X

oAXXRV S` HBMû`Biû /m Tbb;2 ¨ HǶBMiû;`H2- QM

8P2R[X], Z 1

0

P(t)g(t)dt= 0

oAXXkV .ǶT`ĕb H2 i?ûQ`ĕK2 /2 q2B2`bi`bb- BH 2tBbi2 mM2 bmBi2 (Pn) /ǶûHûK2Mib /2R[X] i2HH2 [m2 kPn gk1,[0,1]!0X PM HQ`b

Z 1 0

Png Z 1

0

g²  kPn gk1,[0,1]

Z 1 0 |g|!0 2i +QKK2R1

0 Png2bi iQmDQm`b MmH-

Z 1 0

g²= 0 g² ûiMi +QMiBMm2 2i TQbBiBp2 bm`[0,1]+2+B 2Mi`ŗM2 [m2

8t2[0,1], g(t) = 0

oAX"XRV u 7! e ^xuf(u) ûiMi +QMiBMm2 bm` R<sup>+</sup>- H2 i?ûQ`ĕK2 7QM/K2MiH BM/B[m2 [m2 h 2bi mM2 T`BKBiBp2 /2 +2ii2 7QM+iBQM bm`R⁺X lM2 BMiû;`iBQM T` T`iB2 /QMM2 HQ`b

8b >0, Z b

0

f(t)e ^(x+a)tdt=⇥

h(t)e ^at⇤^b

0+a Z b

0

e ^ath(t)dt

G2 K2K#`2 /2 ;m+?2 /K2i mM2 HBKBi2 Uû;H2 ¨Lf(x+a)V [mM/b!+1U+`x+a2EVX PM /QM+

Lf(x+a) = lim

b!+1

Ç

h(b)e ^ab+a Z b

0

e ^ath(t)dt å

R8

S` BHH2m`b-h/K2i mM2 HBKBi2 }MB2 2M+1U+`x2E ⇢E<sup>0</sup>V 2ie <sup>ab</sup>!0 [mM/b!+1 U+`

a >0VX BMbB-

L(f x+a) = lim

b!+1a Z b

0

e ^ath(t)dt=a Z +1

0

e ^ath(t)dt HǶ2tBbi2M+2 /2 HǶBMiû;`H2 ûiMi +QMbû[m2M+2 /2 HǶ2tBbi2M+2 /2b mi`2b HBKBi2bX

oAX"XkV t7!e ^at 2bi mMC¹ /BzûQKQ`T?BbK2 /2R<sup>+⇤</sup> /Mb]0,1[X PM T2mi BMbB TQb2`u=e ^ta TQm`

Q#i2MB` U+2 [mB BM+Hmi HǶ2tBbi2M+2 /2 HǶBMiû;`H2 /m K2K#`2 /2 /`QBi2V

a Z +1

0

e ^t(n+1)ah(t)dt= Z 1

0

uⁿh

Å ln(u) a

ã du

S` BHH2m`b H2 KK#`2 /2 ;m+?2 pmi<sub>n+1</sub><sup>1</sup> Lf(x+(n+1)a)2i 2bi MmH /ǶT`ĕb H [m2biBQM T`û+û/2Mi2X BMbB

8n2N, Z 1

0

uⁿh

Å ln(u) a

ã du= 0 oAX"XjV G 7QM+iBQM g : u7! hÄ _ln(u)

a

ä 2bi +QMiBMm2 bm` ]0,1]2i T`QHQM;2#H2 T` +QMiBMmBiû 2M 0 U+` h/K2i mM2 HBKBi2 }MB2 2M +1+` x2 EVX p2+ H2b [m2biBQM oAX"XkV 2i oAXXkV- QM 2M /û/mBi [m2g 2bi MmHH2X ZmM/up`B2 /Mb[0,1]- ^ln(u)_a p`B2 /MbR<sup>+</sup>2ih2bi /QM+ MmHH2 bm`R⁺X oAX*X aQBi f i2HH2 [m2 E 2bi MQM pB/2X amTTQbQMb [m2Lf = 0c H [m2biBQM T`û+û/2Mi2 BM/B[m2 [m2 8x2E- mM2 T`BKBiBp2 /2 u7!e ^xuf(u)2bi MmHH2 bm` R<sup>+</sup>X SQm` iQmi x/2E- u7!e ^xuf(u) 2bi /QM+ MmHH2 bm`R<sup>+</sup>X *QKK2EMQM pB/2 U2i +QKK2expM2 bǶMMmH2 Tb bm`RVf 2bi /QM+ MmHH2 bm`

R⁺X

G2 MQvm /2 HǶTTHB+iBQM HBMûB`2L2bi /QM+ `û/mBi ¨{0}2i L2bi BMD2+iBp2X

1im/2 2M H #Q`M2 BM7û`B2m`2 /2 EX

oAAXX *QKK2f 2bi TQbBiBp2- QM E=E⁰ KBb mbbBLf [mB 2bi /û+`QBbbMi2 bm`E U8x, y2E i2Hb [m2xy- 8t 2R⁺ QM f(t)e ^(t)x f(t)e ^(t)y 2i /QM+Lf(x) Lf(y)VX 1M T`iB+mHB2`- T`

i?ûQ`ĕK2 /2 HBKBi2 KQMQiQM2-Lf /K2i /2b HBKBi2b mt #Q`M2b /2 HǶBMi2`pHH2E Uûp2Mim2HH2K2Mi +12M H #Q`M2 BM7û`B2m`2 bB H 7QM+iBQM MǶ2bi Tb KDQ`û2VX

oAAXXRV PM bmTTQb2Lf #Q`Mû2 bm`E 2i QM MQi2M mM KDQ`Mi /2 +2ii2 7QM+iBQMX JQMi`QMb [m2

8b 0, Z b

0

f(t)e ^↵tdtM (⇤)

6BtQMb /QM+b 0cGb : x7!Rb

0f(t)e ^xt dt2bi +QMiBMm2 bm`RT` i?ûQ`ĕK2 bm` H2b BMiû;`H2b ¨ T`Kĕi`2b +`

@ 8x2R, t7!f(t)e ^xt 2bi +QMiBMm2 bm`[0, b]

@ 8t2[0, b], x7!f(t)e ^xt 2bi +QMiBMm2 bm`R

@ 8[u, v]⇢R, 8x2[u, v], 8t2[0, b], |f(t)e ^xt|f(t)e ^ut2i H2 KDQ`Mi 2bi BMiû;`#H2 bm`[u, v]

TmBb[m2 +QMiBMm bm` +2 a1:J1LhVX

P`-8x2E, Gb(x)Lf(x)U+`f 2bi TQbBiBp2V 2i /QM+8x2E, Gb(x)MX 1M 7BbMi i2M/`2x p2`b↵- QM Q#iB2Mi(⇤)X

b 7!Rb

0 f(t)e ^↵t dt 2bi BMbB KDQ`û bm`R⁺ 2i +Ƕ2bi mM2 7QM+iBQM +`QBbbMi2 U+` f 2bi TQbBiBp2VX 1HH2 /K2i /QM+ mM2 HBKBi2 }MB2 [mM/b!+12i /QM+

↵2E⁰ =E

oAAXXkV S` +QMi`TQbû2- bB↵2/EHQ`bLf MǶ2bi Tb #Q`Mû2X p2+ H `2K`[m2 BMBiBH2 /2 KQMQiQMB2- QM /QM+

x!↵lim⁺Lf(x) = +1 Re

oAAX"X PM B+B UTQm` x2E⁰VLf(x) =R+1 0

cos(t) (1+t)^x dtX

oAAX"XRV aB x >1 HQ`b <sub>(1+t)</sub><sup>cos(t)</sup>x  (1+t)<sup>1 x</sup> 2bi BMiû;`#H2 m pQBbBM;2 /2+12ix2EX aBx1HQ`b QM `2K`[m2 [m2

8n2N,

Z (n+1)⇡

n⇡

|cos(t)| (1 +t)^x dt

Z n⇡+⇡/4 n⇡

p dt

2(1 +t)^x

⇡ 4p

2(1 + (n+ 1)⇡)^x PM +QM+Hmi HQ`b +QKK2 2M oX [m2x /2EX BMbB

E=]1,+1[ oAAX"XkV AH bǶ;Bi /2 pQB` bB G : b7!Rb

0 cos(t)

(1+t)^x dt /K2i mM2 HBKBi2 2M+1X

@ aBx >0- mM2 BMiû;`iBQM T` T`iB2b /QMM2

8b 0, G(b) = sin(b) (1 +b)^x +x

Z b 0

sin(t) (1 +t)^x+1 dt

H2b /2mt i2`K2b /m K2K#`2 /2 /`QBi2 /K2ii2Mi mM2 HBKBi2 [mM/b!+1U2M T`iB+mHB2`- H 7QM+iBQM bQmb HǶBMiû;`H2 2bi BMiû;`#H2 +` /QKBMû2 T`1/t<sup>x+1</sup>VX AH 2M 2bi /2 KāK2 /m K2K#`2 /2 ;m+?2 2ix2E⁰X

@ aBx= 0 HQ`bG(b) = sin(b)MǶ/K2i Tb /2 HBKBi2 2M+12i02/E⁰X

@ aBx <0 HQ`bR2n⇡+⇡/4 2n⇡

cos(t)

(1+t)^x dt ^p1₂R2n⇡+⇡/4 2n⇡

1

(1+t)^x dt ₄^p⇡₂(1 +n⇡) ^xM2 i2M/ Tb p2`b 0 [mM/ n!+1X BMbB G(2n⇡+⇡/4) G(2n⇡)M2 i2M/ Tb p2`b 0 2i x /2E⁰ UbBMQM- MQi`2 /Bzû`2M+2 i2M/`Bi p2`b0 +QKK2 /Bzû`2M+2 /2 /2mt i2`K2b vMi H KāK2 HBKBi2VX

PM /QM+ KQMi`û [m2

E⁰=]0,+1[

oAAX"XjV PM `ûmiBHBb2 H2 +H+mH /2 oAX"XkV /Mb H2 +bx >0[mB MQmb /QMM2

8x >0, Lf(x) =x Z +1

0

sin(t) (1 +t)^x+1 dt

1M miBHBbMi H2 i?ûQ`ĕK2 /2 +QMiBMmBiû /2b BMiû;`H2b ¨ T`Kĕi`2- QM Q#iB2Mi [m2x7!R+1 0

sin(t) (1+t)^x+1 dt 2bi +QMiBMm2 bm`[1,+1[UQM miBHBb2 H /QKBMiBQM _(1+t)^sin(t)x+1  (1+t)^{1 2}VX PM 2M /û/mBi [m2

xlim!1⁺Lf(x) = Z +1

0

sin(t) (1 +t)² dt

1M `27BbMi mM2 BMiû;`iBQM T` T`iB2 /Mb HǶmi`2 b2Mb- QM KQMi`2 [m2 Lf(x) ! Lf(1) = R+1

0

cos(t)

1+t dt[mM/x!1⁺X

lM2 miBHBbiBQM /2 H i`Mb7Q`KiBQM LX

oAAAXX aQB2MiP, Q2PXt7!P(t)Q(t)e ^t 2bi +QMiBMm2 bm`R<sup>+</sup>2i /QKBMû2 T` 1/t² m pQBbBM;2 /2 +1U+`QBbbM+2b +QKT`û2bVX *Ƕ2bi /QM+ mM2 7QM+iBQM BMiû;`#H2 bm`R⁺2i bQM BMiû;`#H2 2tBbi2 7Q`iBQ`B bm`R⁺X

oAAAX"X GǶTTHB+iBQM 2bi #B2M /û}MB2- TQbbĕ/2 H bvKûi`B2 ?2`KBiB2MM2 U(P, Q) = (Q, P)V 2i 2bi HBMûB`2 T` `TT`Qi ¨ H b2+QM/2 p`B#H2 UT` HBMû`Biû /m Tbb;2 ¨ HǶBMiû;`H2VX .2 THmb bBP2P- (P, P) =R+1

0 |P(t)|²e ^t dt 0 2i bB +2ii2 [mMiBiû 2bi MmHH2 HQ`bP = 0U+`t 7!|P(t)|²e ^t 2bi HQ`b +QMiBMm2 TQbBiBp2 /ǶBMiû;`H2 MmHH2 2i /QM+ MmHH2 2i HǶ2tTQM2MiB2HH2 M2 bǶMMmH2 TbVX

PM }MH2K2Mi mM T`Q/mBi b+HB`2 bm` H2C@2bT+2 p2+iQ`B2HPX

Rd

oAAAX*X U 2bi HBMûB`2 T` HBMû`Biû /m Tbb;2 ¨ H /û`Bpû2X .2 THmb-U(X⁰) = 02i 8n2N^⇤n U(Xⁿ)(t) =e^tD(ntⁿe ^t) = ntⁿ+n²t^{n 1}

BMbB- 8n 2N, U(ⁿ)2 PX *QKK2 iQmi ûHûK2Mi /2 P 2bi +QK#BMBbQM HBMûB`2 /ǶûHûK2Mib /2 H 7KBHH2(X<sup>n</sup>)n2N- P 2bi bi#H2 T`UX 6BMH2K2Mi

U 2L(P)

oAAAX.X PM U(P)(t)Q(t)e ^t=D(te ^tP⁰(t))Q(t)X lM2 BMiû;`iBQM T` T`iB2b /QMM2

8a 0, Z a

0

U(P)(t)Q(t)e ^tdt=î

te ^tP⁰(t)Q(t)óa 0

Z a 0

te ^tP⁰(t)Q⁰(t)dt

S` +`QBbbM+2b +QKT`û2b- H2b /Bzû`2Mib i2`K2b /K2ii2Mi mM2 HBKBi2 [mM/a!+12i QM Q#iB2Mi Z +1

0

U(P)(t)Q(t)e ^tdt=

Z +1 0

te ^tP⁰(t)Q⁰(t)dt PM KQMi`2 /2 KāK2 [m2

Z +1 0

P(t)U(Q)(t)e ^tdt=

Z +1 0

te ^tP(t)⁰Q⁰(t)dt

G /û`Bpû2 /m +QMDm;mû ûiMi H2 +QMDm;mû /2 H /û`Bpû2 QM HǶû;HBiû /2b i2`K2b T`û+û/2Mib , (U(P), Q) = (P, U(Q))

oAAAX1X U(1) = 0 KQMi`2 [m2U /K2i m KQBMb0+QKK2 pH2m` T`QT`2X aQBi mM2 pH2m` T`QT`2 /2U 2i P mM p2+i2m` T`QT`2 bbQ+BûX PM HQ`b

kPk²= ( P, P) = (U(P), P) = (P, U(P)) = (P, P) = kPk²

2i +QKK2P 6= 0U+Ƕ2bi mM p2+i2m` T`QT`2V QM kPk<sup>2</sup>6= 02i /QM+ = +Ƕ2bi ¨ /B`2 2RX aQB2Mi , µ /2mt pH2m`b T`QT`2b /BbiBM+i2b U`û2HH2b /ǶT`ĕb +2 [mB T`û+ĕ/2V 2i P, Q /2b p2+i2m`

T`QT`2b bbQ+BûbX PM

(P, Q) = (P, Q) = (U(P), Q) = (P, U(Q)) =µ(P, Q)

*QKK2 6=µ- QM /QM+(P, Q) = 02i H2b p2+i2m`b T`QT`2b bbQ+Bûb ¨ /2b pH2m`b T`QT`2b /BbiBM+i2b bQMi Q`i?Q;QMmt UH2b bQmb@2bT+2b T`QT`2b bQMi /2mt ¨ /2mt Q`i?Q;QMmtVX

oAAAX6XRV U(P)(t) = e^tD(te ^tP⁰(t)) = tP⁰(t) +P⁰(t) +tP⁰⁰(t)X aB U(P) = P-P 2bi bQHmiBQM /2 HǶû[miBQM /Bzû`2MiB2HH2

ty⁰⁰(t) + (1 t)y⁰(t) y(t) = 0

oAAAX6XkV aQBi n H2 /2;`û /2 PX AH 2tBbi2 Q 2 Cn 1[X] 2i a 2 C^⇤ i2Hb [m2 P(t) = atⁿ +Q(t)X G2 +Q2{+B2Mi /2Xⁿ /MbXP⁰⁰+ (1 X)P⁰ P 2bi na aX PM 2M /û/mBi [m2 = deg(P)X oAAAX:XRV aQBiP 2Pc QM L(XP⁰)(x) =R+1

0 tP⁰(t)e ^tx dtX lM2 BMiû;`iBQM T` T`iB2b U/QMi QM M2 /ûiBHH2 Tb H2 +H+mH p2+ H2 Tbb;2 T` mM2 #Q`M2 ~QiiMi2V /QMM2 U+QKTi2@i2Mm /2b +`QBbbM+2b +QKT`û2b 2i /2 H [m2biBQM AoXV

8x >0, L(XP⁰)(x) =

Z +1 0

P(t)(e ^tx xte ^tx)dt= LP(x) x(LP)⁰(x)

R3

2i /2 7ÏQM bBKBHB`2-

8x >0, L(XP⁰⁰)(x) =

Z +1 0

P⁰(t)(e ^tx xte ^tx)dt

= L(P⁰)(x) +x Z +1

0

tP⁰(t)e ^txdt

= L(P⁰)(x) x Z +1

0

P(t)(e ^tx xte ^tx)dt

= L(P⁰)(x) xLP(x) x²(LP)⁰(x)

amTTbQMbXP⁰⁰+ (1 X)P⁰+nP = 0X PM HQ`bL(XP⁰⁰) +L(P⁰) L(XP⁰) +nL(P) = 0+2 [mB /QMM2

8x >0, x(1 x)(LP)⁰(x) + (1 x)LP(x) +nLP(x) = 0 Q=LP 2bi /QM+ bQHmiBQM bm`]0,+1[/2

x(1 x)y⁰(x) + (n+ 1 x)y(x) = 0 (E_n⁰) oAAAX:XkV am`]1,+1[- HǶû[miBQM(E<sub>n</sub><sup>0</sup>)2bi `ûbQHm2 2i ¨ +Q2{+B2Mib +QMiBMmbX GǶ2Mb2K#H2 /2 b2b bQHm@

iBQMb 2bi Ui?ûQ`ĕK2 /2 *m+?v@GBTb+?BixV mM 2bT+2 p2+iQ`B2H UHǶû[miBQM 2bi ?QKQ;ĕM2V /2 /BK2M@

bBQM1X *QKK2

8x >1, x n 1

x(1 x) = n+ 1

x + n

1 x

mM2 T`BKBiBp2 bm`]1,+1[/2x7!^{x n}x(1 x)¹ 2bix7!lnÄ_{(x 1)}n

xⁿ⁺¹

äX G2 +Qm`b BM/B[m2 [m2 HǶ2Mb2K#H2 /2b bQHmiBQMb bm`]1,+1[/2(E_n⁰)2bi HǶ2bT+2 p2+iQ`B2H 2M;2M/`û T`

fn : x7! (x 1)ⁿ xⁿ⁺¹

_BbQMMQMb T` +QM/BiBQMb Mû+2bbB`2b TmBb bm{bMi2b TQm` i`Qmp2` H2b ûHûK2Mib T`QT`2b /2UX

@ aQBi mM2 pH2m` T`QT`2 2iP mM p2+i2m` T`QT`2 bbQ+BûX AH 2tBbi2n2Ni2H [m2 = n2i P 2bi bQHmiBQM /2 (En) U[m2biBQM oAAAX6VX LP 2bi BMbB bQHmiBQM /2 (E_n⁰) bm` ]1,+1[ 2i LP 2bi KmHiBTH2 /2 fnX S` BHH2m`b

fn(x) = Xn

k=0

( 1)^k k!

Çn k

å k!

x^k+1

p2+ H [m2biBQM AoX" /Mb H2 +ba= 0- QM pQBi [m2fn2bi BK;2 T`L/2Qn=Pn k=0

( 1)^k k!

n k X^kX BMbB- p2+ H T`iB2 oA-P 2bi mM KmHiBTH2 /2Qn UTmBb[m2 LP 2bi KmHiBTH2 /2LQnVX

@ _û+BT`Q[m2K2Mi QM KQMi`2 [m2U Qn= nQn T` mM +H+mH [m2 MQmb QK2iiQMb 2M +2ii2 }M /2 T`Q#HĕK2X

PM /QM+ KQMi`û [m2 H2b pH2m`b T`QT`2b /2U bQMi H2b ûHûK2Mib /2Z 2i [m2 +?[m2 bQmb@2bT+2 T`QT`2 2bi mM2 /`QBi2 p2+iQ`B2HH2 /QMi QM /QMMû mM2 #b2X

oAAAX:XjV .ǶT`ĕb H 7Q`KmH2 /2 G2B#MBx- QM

Pn(t) = e^t Xn

k=0

Çn k å

D^k(e ^t)D^{n k}(tⁿ)

= e^t Xn

k=0

Çn k å

( 1)^ke ^tn!

k!t^k

= n!Qn(t)

Pn 2M;2M/`2 /QM+ mbbB H /`QBi2 p2+iQ`B2HH2 T`QT`2 bbQ+Bû2 ¨ H pH2m` T`QT`2 nX RN