defined on a domain can be expanded into the Taylor series around a point a except a singular point. Also, f( z)

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1 08 Tylo eie nd Mcluin eie A holomophic function f( z) defined on domin cn be expnded into the Tylo eie ound point except ingul point. Alo, f( z) cn be expnded into the Mcluin eie in the open dik with diu fom the oigin O to the neet ingulity. When the two convegence cicle he the oigin O, the Tylo eie eult in the Mcluin eie. At thi time, numeou eqution e geneted between the coefficient of both. 8. Ce without ingul point Theoem 8.. When function f( z) i holomophic in the whole domin D, the following expeion hold fo ny D. f () () f () () 0 (-) - C -! fo 0,,, Poof The function f( z) cn be expnded into the Mcluin eie in D, nd the diu of convegence i R m. Alo, the function f( z) cn be expnded into the Tylo eie ound ny D follow. f() z 0 f () () ( z-)! And thi diu of convegence i lo R t. Hee, ( z-) ( -) - C - z Uing thi, 0 f () () ( z-)! 0 0 f () ()! Since thi h to be equl to the Mcluin eie in D, 0 f () () (-) - C - z 0 (-) - C - z f () () (-) - C - z f () () 0 z! Futhemoe, ince both eie convege unifomly in D, thee hve to be me. If not o, it contdict to the uniquene of the eie. Theefoe, f () () Exmple f() z coz - - C - f () () 0! The Mcluin eie nd the Tylo eie ound e follow. z f m () z co 0! ( z-) f t () z co 0! fo ny D 0,,, - -

2 0 (-) - co C - z Thee diu of convegence e R m R t. Then, fo ny, the following hold. (-) - co C - co! fo 0,,, In fct, when c t (, ) denote the left ide nd c m () denote the ight ide, ech vlue of c t ( 00,) t, -e, i e follow. Moeove, the following how tht the vlue of both ide e equl fo 50,5 t -i. Specil vlue When 0, (-) co fo ny e.g. 0 co in co in !!!! 0! 4 0! 4 -! 4 -! 4 -- When, (-) - - ( - )! co 0 fo ny e.g. co in co 4 in in!!! 4!! 4! 4 -! 4-4! Exmple inhz The Mcluin eie nd the Tylo eie ound e follow. - -

3 f m () z 0 f t () z 0 Fom thee, When 0, -( -) - z! e e - ( z-)! e - (-) - e - (-) - C - (-) -( -) -! e - (-) - e - 0 fo ny fo 0,,, 0 inh coh inh coh !!!! When (-) e - (-) -- e - fo ny 0 coh inh coh inh !!!! Exmple ( z) e z The Mcluin eie nd the Tylo eie ound e follow. f m () z 0 f t () z e 0 Fom thee, e When 0, z! ( z-)! (-) - C -! ( -) e fo ny ( -) 0 0! When, ( -)! ( -)! ( -) e fo ny fo 0,,, ( 4-) e! - -

4 ( -) 0 0! Genelly, ( -)! ( 4-)! ( b - ) be fo ny,b ( b -) 0 0! ( b - )! ( b - )! ( 5-) e! ( b - ) be! In fct, clcultion eult giving viou vlue to b, e follow

5 8. Ce with ingul point Theoem 8.. When function f( z) i holomophic in domin D except ingul point p, the following expeion hold fo.t. < p - f () () f () () 0 (-) - C - fo 0,,,! Whee, the ingul point p i umed to be neet to the oigin O nd the point. Poof The function f( z) cn be expnded into the Mcluin eie in cicle C m of the diu R m p follow. f () () 0 f() z z z < R 0! m p Alo, the function f( z) cn be expnded into the Tylo eie ound except p follow. f() z 0 f () () ( z-)! z- < R t p - The cente of the convegence cicle C t i, nd the diu i R t p -. Hee, ( z-) ( -) - C - z. Uing thi, 0 f () () ( z-)! 0 f () () (-) - C - z Since thi h to be equl to the Mcluin eie in C m C t, 0 f () () (-) - C - z 0 f () () 0 z! When < p -, ince the convegence cicle C t of the Tylo eie include the oigin O, cicle C c ound the oigin O exit touching the cicle C t intenlly. (The diu i p - -. ) Since both eie convege unifomly in C c, thee hve to be me. If not o, it contdict to the uniquene of the eie. Theefoe, - 5 -

6 f () () (-) - C - f ( ) () 0! < p - 0,,, Condition fo the theoem Let i, p piq. Then < p - educe to the following expeion. p( p- ) q( q- ) > 0 Epecilly, when q 0, if p > 0 then < p/ if p < 0 then > p/ Epecilly, when p 0, if q > 0 then < p/ if q < 0 then > p/ - 6 -

7 8. Exmple of Ce : C m C t Thi i ce whee the convegence cicle C m of the Mcluin eie include the convegence cicle C t of the Tylo eie. We conide the following function the exmple. f() z tnh - z The highe ode deivtive i given by the following expeion. (See " 岩波数学公式 Ⅰ" p6,) n tnh - z () The Mcluin eie i f m () z 0 ( n -)! () 0 z! f () Fom thi nd the bove, f m () z tnh - 0 ( -z) n f () 0 () 0 z 0 0! ( -) - z On the othe hnd, the Tylo eie ound i f t () z 0 f () () (-) - C - z (-) n- ( z) n n,,, f () () 0 z! f () () (-) C 0 z 0 f () () (-) - C - z f () 0 f () () () (-) f () () (-) - C - z Fom thi nd the bove, f t () z tnh - - (-) - ( -) (-) - ( -) Then, thee coefficient c m (), c t, e follow. c m tnh () ( -) - > 0 c t (, ) tnh - - ( -) - - ( -) - - C - z C - > 0 (.m) (.t) Since the f( z) h the ingul point t z, the convegence diu of the Mcluin eie i R m nd the convegence diu of the Tylo eie i R t -. Theefoe, i f Re( ) < /, < -. Fo exmple, if /5 0.4 then <

8 Theefoe, the following eqution hve to hold. (-) ( -) - - ( -) tnh tnh C - ( -) -,,, In fct, when /5, the eult tht the eie of the left ide w clculted to the 0000 th tem fo 0 nd,,4 e follow. We cn ee tht even numbe tem of the Tylo eie e lmot 0. Specil vlue We obtin the following fomul fom the vobe exmple

9 Fomul 8.. tnh Re < Poof Fom (.m), (.t) t 0, tnh - tnh ( -) Hee, eplcing with -, tnh - (-) tnh ( -) Since thi function i n odd function, tnh - (-z) -tnh - z. Theefoe, Exmple tnh - tnh - tnh Re Re Re < < <

10 8.4 Exmple of Ce : C m C t Thi i ce whee the convegence cicle C m of the Mcluin eie i included in the convegence cicle C t of the Tylo eie. We conide the following function the exmple. 5 f() z 5-z The Mcluin eie nd the Tylo eie ound e follow. f m () z 0 f t () z 0 z ( z-) 5 0 ( 5-) Then, thee coefficient c m (), c t c m () 5 (-) - C - z, e follow. (.m) c t (, ) 5 ( 5-) (-) - C - (.t) Since the f( z) h the ingul point t z 5, the convegence diu of the Mcluin eie i R m 5 nd the convegence diu of the Tylo eie i R t 5-. Theefoe, i f Re( ) <5/, < p -. Fo exmple, if -7 then < 5-. Theefoe, the following eqution h to hold. 5 ( 5-) (-) - C - 5 Re( ) < 5/ 0,,, In fct, when -7, the eult tht the eie of the left ide w clculted to the 00 th tem fo 0,,,7 e follow. We cn ee tht coefficient of the Tylo eie nd the Mcluin eie e conitent

11 Geometic Seie with coefficient Fom the vobe exmple, we obtin the following fomul. Fomul 8.4. (-) - C x Epecilly, when, x x ( -x) x - -x -x x > x > 0,,, Poof Replcing the convegence diu 5 with in the vobe exmple, we obtin (-) - C - ( -) Tnfom thi follow. Hee, let Then C - x -x x, - -x Re( ) < / 0,, - ( -) Re( ) < / 0,,, Subtituting thee fo thevobe, (-) - C - -x x A long </, x > i gunteed. x -x x > 0,,, educe When 0, x - -x x Thi i the uul geometic eie. x > - -

12 When, x x ( -x) Thi i the geometic eie with coefficient. Exmple x > Exmple Exmple ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) 4 i i i - ( i) - -

13 8.5 Exmple of Ce : C m C t Thi i ce whee the convegence cicle C m of the Mcluin eie nd the convegence cicle C t of the Tylo eie e ovelpping ptilly. We conide the following function the exmple. f() z z The highe ode deivtive i given by the following expeion. (See " 岩波数学公式 Ⅰ" p,) z () n Subtituting thi fo Theoem 8.., c m () c t (, ) n!( -) n z - n in( n ) cot - z 0 - f () () 0!( -)!! c m () (-) in (-) in cot - 0 f () () (-) - C - ( -) - in cot - 0 Re() z 0 in cot - ( -) - C - c t (, ) (-) - in cot - ( -) - C - (.t) (.m) Since the f( z) h the ingul point t z i, the convegence diu of the Mcluin eie i R m nd the convegence diu of the Tylo eie i R t i -. Theefoe, i f Im( ) < /, < i -. Fo exmple, if 0.7 then < i -. Theefoe, the following eqution h to hold. (-) - in cot - ( -) - C - (-) in - -

14 In fct, when, the eult tht the eie of the left ide w clculted to the 0000 th tem fo 0,,, e follow. We cn ee tht odd numbe tem of the Tylo eie e lmot 0. Altenting eie of powe of / Fom the vobe exmple, we obtin the following fomul. Fomul Poof Subtituting / fo (.t), c t, - (-) in (-) in - in ( ) cot C (-) - C (-) in Hee, uing Diichlet chcte ( m, j) Then (-) in c t, (,) c m () (-) in 4 - (-) - m: modulu, j: index, 4 (-) - C (,) - - C C

15 When 0, c t 0, c m () 0 (-) 0 in Fom thee, 0 4 (-) -0 C 0 (,) ( 0) (-) 4 (,) When, c t, c m () (-) in Fom thee, (-) (,) 4 (-) - C (,) Adding 0/ 0 to the both ide, we obtin the deied expeion

16 8.6 Sum of Stieltje contnt In thi ection, we pove the fomul of um of Stieltje contnt. It eem tht thi fomul w dicoveed by O. Michev by 008. ( ). Although I do not know how D.Michev poved thi, the poof pefomed in thi ection i vey good exmple of ppliction of the Theoem 8... Fomul 8.6. (O. Michev) When e Stieltje contnt nd () n () function, The following expeion hold. 0 i the n-th ode diffeentil coefficient of Riemnn zet n (-) n n! () n () 0 n 0,,, (.n) Poof It i known tht the function f( z ) ( z -) ( z ) i expnded to Tylo eie follow. ( z-) () z (-) - - ( z-) : Stieltje contnt Then f () () 0 (-) - - > 0 Next, the Mcluin expnion of f( z ) i follow. f () () 0 ( z-) () z z 0! Since f( z ) i poduct of two function, the highe ode deivtive i given by the following Leibniz ule. n () n n g() z h() z 0 g ( n- ) () z h () () z Hee, put g() z z-, h() z () z, then g () 0 () z z-, g () () z, g () () z 0 (,,4,) Subtituting thee fo the bove, f () n () z ( z-) () z () n f () n () n () z ( z-) () z The diffeentil coefficient t z 0 e f () () 0 ( - ) () 0- () Then f () () ( - ) () 0 0- () () 0!! n n - ( n- ) () n ( n- ) () z ( z-) () n () z () 0 0,,, 0,,, n z n ( z- ) () n () z Now, the function f( z ) ( z -) ( z ) i holomophic on the whole complex plne. So, Theoem 8.. hold nd the following eltion exit between the (.t) nd (.m). (.t) (.m) - 6 -

17 f ( ) () (-) - C - f ( ) () 0! fo 0,,, Then, ubtituting (.t) nd (.m) fo the both ide one by one ( 0,,, ), it i follow. When 0, Left: Right: Fom thee, f () () (-) -0 C f () () (-) C 0 - f () 0 () () 0 () 0 0! 0! (-) - - (-) -0 C ( - )! - 0! () 0 () 0 (.0) When, Futhe, Uing thi, Left: Right: Fom thee, f () () (-) - C - (-) - - -!( - )!! - -!!( - )! 0 0 0!!!! 0!!! f () () (-) - C -! f () () 0 () 0 () 0- () () 0!! () 0 () 0- () () 0 Subtituting (.0) fo thi,! () 0 () 0 () 0 () 0- () () 0-7 -

18 Fom thi, -! () () 0 (.) When, Futhe, Uing thi, Left: Right: Fom thee, f () () (-) - C - - (-) !!( - )! -!!( - )! !!!! 0!!! f () () (-) - C - -! f () () 0 () () 0- () () 0!! - Subtituting (.) fo thi, Fom thi, () () 0- () () 0 -! () () - 0 () () 0- () () 0 -! () () 0 (.) When, Futhe, f () () (-) - C -! (-) !( - )!!!( - )! - 8 -

19 Uing thi, Left: Right: Fom thee, f () () (-) - C -! f () () 0 () () 0- () () 0!! () () 0- () () 0 Subtituting (.) fo thi, Fom thi,! () () 0 () () 0- () () 0 -! () () 0 (.) Heefte, by induction, we obtin the deied expeion. Specil vlue The following pecil vlue e known bout Riemnn zet function. Theefoe, () 0 () 0 - ( 0 ), () () ! () 0 () 0 log Glihe-Kinkelin contnt -! () () 0 log Renewed Added Section 6. Alien' Mthemtic Kno. Kono - 9 -

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