數uni5B78傳uni64AD 31uni53772uni671F, pp. 76-80uni52D2uni8C9Duni683Cuni55AE調收uni6582uni5B9Auni7406uni4E00個有趣uni7684應uni7528uni8B49明uni5C24uni62C9uni7684uni53CDuni6B63切公式uni9EC3見uni5229uni4E00. uni524D言uni5C24uni62C9 (Euler) 著uni540Duni7684uni53CDuni6B63切公式tan1 x =summationdisplayn=022n(n!)2(2n + 1)!x2n+1(1 + x2)n+1, (1)uni767Cuni73FEuni57281755年, uni5728計算uni5713uni5468率uni7684數值上是uni4E00uni689Duni975E常uni91CD要而且很有效率uni7684uni53CDuni6B63切級數。不像uni683Cuni96F7uni54E5uni91CC (Gregory) uni7684uni53CDuni6B63切級數uni90A3uni6A23, uni5B83uni5C0D所有uni7684 x 值uni90FD是收uni6582uni7684, uni7279uni5225是uni8F03uni5C0Funi7684角度。uni8209個例uni5B50來說, uni5229uni7528uni5C24uni62C9於1779年出版uni5728 Nova Acta Petropolitanae uni7684uni53CDuni6B63切型公式pi4 = 5tan1parenleftBig17parenrightBig+ 2tan1parenleftBig 379parenrightBig, (2)uni914Duni5408公式 (1), 我們得uni5230下列uni9AD8收uni6582級數pi4 =710braceleftBig1 + 23parenleftBig 2102parenrightBig+ 2 43 5parenleftBig 2102parenrightBig2+ bracerightBig+7584105braceleftBig1 + 23parenleftBig144105parenrightBig+ 2 43 5parenleftBig144105parenrightBig2+ bracerightBiguni5229uni7528uni9019uni689D級數, uni5C24uni62C9uni5728uni4E00個uni5C0F時內算出了uni4E8C十位uni5713uni5468率uni7684uni5C0F數!uni4E8C. 舊uni7684uni8B49明uni6CD5uni9996先, 我們uni5B9A義超幾何級數uni5982下 1:uni5B9A義 I: 級數summationdisplayn=0tn 被稱uni70BA超幾何級數, 假使 tn+1tnuni70BA n uni7684有uni7406uni51FD數。76uni52D2uni8C9Duni683Cuni55AE調收uni6582uni5B9Auni7406uni4E00個有趣uni7684應uni7528 77uni5B9A義 II: ()n = ( + 1)( + 2)(d + n 1) =n1productdisplayj=0 + j, negationslash= 0。我們稱 ()nuni70BAuni6CE2伽瑪 (Pochhammer) uni7B26號。uni7279uni5225uni5730, ()0 = 1, (1)n = n!。uni7531於 tn+1tnuni70BA n uni7684有uni7406uni51FD數, uni5728不uni5931uni70BAuni4E00般性uni7684情uni6CC1下, uni53EF以uni5C07uni5B83uni5BEB成下列形式:R(n) = (n + a1)(n + a2)(n + ap)(n + 1)(n + b1)(n + b2)(n + bq)uni56E0uni6B64, 我們uni5C31有了超幾何級數uni6A19uni6E96uni7B26號uni7684uni5B9A義:uni5B9A義 III: pFqbracketleftBigga1,a2,.,apb1,b2,.,bq ; zbracketrightBigg=summationdisplayn=0(a1)n(a2)n (ap)nn!(b1)n(b2)n (bq)nzn.uni5229uni7528uni6B64uni7B26號, 我們列出uni4E00些熟悉uni7684級數來:ez =summationdisplayn=0znn! = 0F0bracketleftBigg; zbracketrightBiggcosz =summationdisplayn=0(1)nz2n(2n)! = 0F112;z24sinz =summationdisplayn=0(1)nz2n+1(2n + 1)! = z0F132;z24sin1 z = z2F112,1212; z2, |z| q + 1 時, uni5C0D所有uni7684 z negationslash= 0 uni7686uni70BAuni767Cuni6563。3. p = q + 1 時, uni5C0D |z| 1 uni70BAuni767Cuni6563。78 數uni5B78傳uni64AD 31uni53772uni671F uni6C1196年 7月5. p = q + 1 時, uni5C0D |z| = 1 uni70BA收uni6582或uni767Cuni6563uni5247uni672Auni78BAuni5B9A。其中, uni7279uni5225著uni540Duni7684例uni5B50uni70BA 2F1bracketleftBigga,bc ; zbracketrightBigg。 偉uni5927uni7684uni9AD8斯 (Gauss) 曾uni5C0Duni6B64做uni904E廣uni6CDBuni7684uni7814究並uni57281812年uni767C表uni6B64uni7814究uni7684uni6F14uni8B1B。接著, 我們先列出uni683Cuni96F7uni54E5uni91CCuni57281671年uni767Cuni73FEuni7684uni53CDuni6B63切級數tan1 x =summationdisplayn=1(1)nx2n+12n + 1 , |x| 1. (3)uni7136後, uni5229uni7528uni6CE2伽瑪uni7B26號 ()n uni548C恆等式 (32)n(12)n = 2n + 1, 我們uni5C07公式 (3) uni5BEB成超幾何級數uni7684形式:tan1 x = x2F11,1232;x2.uni73FEuni5728考慮下列uni7531uni9AD8斯uni7684uni535A士論文指uni5C0E教uni6388普uni6CD5uni592B(Pfaff) 所uni767Cuni73FEuni7684uni8B8Auni63DB公式 22F1a,bc; z = (1 z)a 2F1a,cbc; z1 z, |z| 1 且 | z1z| 1.設 a = 1, b = 12, c = 32, z = x2 我們uni5C31uni53EF得uni5230tan1 x = x1 + x2 2F11,132; x21 + x2uni7531於 (2n + 1)! = (2)2n = 22nn!(32)2n, 上述方程式uni5C31給出了tan1 x =summationdisplayn=022n(n!)2(2n + 1)!x2n+1(1 + x2)n+1.uni4E09. 新uni7684uni8B49明uni6CD5uni975E常感uni8B1Duni73FE代強而有uni529Buni7684數uni5B78分析工具! 下uni9762我們uni5C07經uni7531uni7279殊uni7684積分技uni5DE7, 再uni5229uni7528uni52D2uni8C9Duni683C(Lebesgue) uni7121uni7AAE級數uni55AE調收uni6582uni5B9Auni7406來uni8B49明公式 (1)。考慮下列式uni5B50integraldisplay 10x1 + 2x2d =bracketleftBigtan1 xbracketrightBig10= tan1 x.uni52D2uni8C9Duni683Cuni55AE調收uni6582uni5B9Auni7406uni4E00個有趣uni7684應uni7528 79經uni7531 = cos uni7684替代, 我們uni53EF以得uni5230tan1 x =integraldisplay pi/20xsin1 + x2 cos2 d.uni73FEuni5728, 我們先使uni7528下列uni95DCuni9375性uni7684技uni5DE7:1 + x2 cos2 = 1 + x2(1 sin2 ) = 1 + x2 x2 sin2 = (1 + x2)parenleftBig1 + x2 x2 sin2 1 + x2parenrightBig= (1 + x2)parenleftBig1 x2 sin2 1 + x2parenrightBig.uni5247tan1 x =integraldisplay pi/20xsin1 + x211 x2 sin2 1 + x2d.接著uni5229uni7528幾何級數11 u = 1 + u + u2 + , |u| 1,我們uni64F4充積分uni9805中第uni4E8C個分式uni5C31得uni5230了tan1 x =integraldisplay pi/20bracketleftBigg summationdisplayn=0x2n+1(1 + x2)n+1 sin2n+1 bracketrightBiggd. (4)接下來, 應該是使uni7528uni73FE代強而有uni529Buni7684uni91CD型武uni5668uni7684時候了! 下述uni5B9Auni7406uni5C31是uni9019uni9805裝備。uni52D2uni8C9Duni683Cuni7121uni7AAE級數uni55AE調收uni6582uni5B9Auni7406 3: 設 cn(x) 0, uni5247integraldisplayEbracketleftBig summationdisplayn=1cn(x)bracketrightBigdx =summationdisplayn=1integraldisplayEcn(x)dx,uni53EA要等號兩邊uni7686uni70BA收uni6582。uni73FEuni5728, 使uni7528uni53E6uni4E00個uni95DCuni9375性un