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½Ö³£ª¾¹D¤û¹y¡]1642¡ã1727¦~¡^¬O·L¿n¤À³Ì­«­nªº½l³yªÌ¡A¦ý«e¦³¥j¤H¡A«á¦³¨ÓªÌ¡A¤û¹y¦¡ªº·L¿n¤À¨ì©³¬O¤°»ò¼Ë¤l«o¬OÄÇ´I½ì¨ýªº¤@­Ó°ÝÃD¡C ¿n¤ÀªºÆ[©À¥i»··¹¨ìªü°ò¦Ì¼w¡C»·ªº¤£»¡¡A±q¤Q¤C¥@¬öªì´Á¨ì¤û¹y¶i¤J·L¿n¤À¾ú¥v¤§«e¡AÁÙ¦³ Fermat¡BWallis µ¥¤H°µ¤F¨Ç·L¿n¤À¾Çªº¶}®i¤u§@¡C ©M¤û¹y¦P®É¤Îµy«áªº¡AÁÙ¦³µÜ¥¬¥§¯÷¡B²Ä¤@¥Nªº Bernoulli ¥@®aµ¥·L¿n¤À¤Hª«¡C¤û¹y¥H«á¡A·L¿n¤ÀÄ~Äòµo®i¡A»â°ìÂX±i¤F¡C¥~»ª¤]ÅܤF¡A¤@ª½¨ì¤@¦Ê¤­¤Q¦~«áªº¤Q¤E¥@¬ö¤U¥b¸­¤~©w«¬¡C¦pªG¤û¹y¦A¥Í¡A®³°_²{¥Nªº·L¿n¤À½Ò¥»¡A¥L¤@©w­n¦n¤@°}¤l¤~·|²ßºD©ó½Ò¥»ªºªí²{¤è¦¡¡F°Ñ¥[·L¿n¤À¦Ò¸Õ¡A¤]¥i¯à¦³¦n´XÃDÃD¥Ø¬Ý¤£À´¡C

¤û¹y¦b·L¿n¤À¤è­±ªº²Ä¤@¥ó­«¤jµo²{´N¬O¤G¶µ®i¶}¦¡¡A¦Ó¥B¤]´N¬O¤G¶µ®i¶}¦¡¨Ï¥L¦b·L¿n¤À¤è­±¦³¤F­«¤jªº¬ð¯}¡C

¥¿¾ã¼Æ«ü¼Æªº¤G¶µ®i¶}¦¡

\begin{displaymath}
(1+x)^n = 1+ {n \choose 1} x + {n \choose 2} x^2 + \cdots
+ {n \choose n} x^n
\end{displaymath}

¦­¦b¤û¹y¤§«e´Nª¾¹D¤F¡A¥Î¥¦¨Ó¨D f(x) = xn¡Aªº·L¤À¤]¬O«e¤H´N¤w¸gª¾¹Dªº¨Æ¡]¥Î²{¥Nªº²Å¸¹¡^
f'(x) = $\displaystyle \lim_{h \rightarrow 0}\frac{(x+h)^n-x^n}{h}=\lim_{h \rightarrow 0}\frac{x^n(1+\frac{h}{X})^n-x^n}{h}$  
  = $\displaystyle \lim_{h \rightarrow 0}
\frac{x^n(1+\frac{nh}{x}+h^2(\cdots))^n-x^n}{h}$  
  = $\displaystyle \lim_{h \rightarrow 0}\frac{nhx^{n-1}+h^2(\cdots)}{h}$  
  = $\displaystyle \lim_{h \rightarrow 0}nx^{n-1}+h(\cdots) = nx^{n-1}$ (1)

1665¦~¡A¤û¹yµo²{¤F¤@¯ë«ü¼Æªº¤G¶µ¾­¯Å®i¶}¦¡¡G
$\displaystyle (1+x)^{\alpha}$ = $\displaystyle 1+ {n \choose 1} x + {n \choose 2} x^2+ \cdots + {\alpha \choose k} x^k + \cdots$ (2)
$\displaystyle {\alpha \choose k}$ = $\displaystyle \frac{\alpha(\alpha -1) \cdots (\alpha -k+1)}{k!}$ (3)

¦]¦¹ $f(x) = x^{\alpha}$ ªº·L¤À¤]¥i¨Ì¼Ëµe¸¬Äª¡A»´©ö¨D±o¡FÁöµM n ´«¦¨ £\ «á¡A (1)¦¡¤¤ªº (¡K) Åܦ¨µL½a¶µ¡A¦ý¤û¹y¬O¤£¦b¥Gªº¡C

¥Ñ (1+x)n¡A¦ü¥G¥u¬O§â ${n \choose k}$ Åܦ¨µL½a¶µ¡A¦ý¨Æ¹ê¤W¡A ·í®É¹ï¤G¶µ«Y¼Æªº¤F¸Ñ¨Ã¤£¬O¥¦ªº¤½¦¡¡]·í®É¨S¦³³o¼Ëªº¤½¦¡¡^¡A ¦Ó¬O¥¦¦b Pascal ¤T¨¤§Î¤¤¬°¤W­±¨â¶µ¤§©M³o¼ËªºÃö«Y¡C ¤û¹y´N¬O±q³o¼ËªºÃö«Y¡A¸g¹L¤¾ªøªº¤º´¡±Àºt¤u§@¡A¦Ó²q¥X¤½¦¡(2)©M(3)¡C

¤û¹y¦^¾Ð»¡¡G¡u¦b1664»P1665¦~¶¡ªº¥V¤Ñ¡A§ÚŪ¤F Wallis ªº¡mArithmetica Infinitorum¡n¡A¤]·Q¥Î¥¦ªº¤èªk¨Ó´M§ä¶êªº­±¿n¡A§Úµo²{¤@­Ó¨D¶ê­±¿nªºµL½a¯Å¼Æ¡A¥H¤Î¥t¤@­Ó¨DÂù¦±½u­±¿nªºµL½a¯Å¼Æ¡K¡K¡C¡v

Wallis ª¾¹D

\begin{displaymath}
\frac{\pi}{4} = \int^1_0(1-x^2)^{\frac{1}{2}}dx
\end{displaymath} (4)

¦Ó¬°¤F¨D±o¥kÃ䪺¿n¤À¡A¥LÅý p, q ¦b¥¿¾ã¼Æ¤¤ÅܤơA­pºâ

\begin{displaymath}
a_{p,q} = \int^1_0(1-x^{\frac{1}{q}})^p dx
\end{displaymath}

¤§­È¡C¥L§ä¨ì ap,q ¤§¶¡ªº¤@¨ÇÃö«Y¡AµM«á¥Î¤º´¡ªk¡A§â³o¨ÇÃö«Y±Àºt¦Ó±o¤@¨Ç p¡Bq ¤£¬O¾ã¼Æ®Éªº ap,q ¤§­È¡C ¦A¸g¹L«D±`½ÆÂøªº¹Lµ{¡A¥L²×©ó±À±o
\begin{displaymath}
\frac{\pi}{4} = a_{\frac{1}{2},\,\frac{1}{2}} = \frac{2 \cdo...
...ot \cdots \cdot
\frac{2n \cdot (2n+2)}{(2n+2)^2} \cdots\cdots
\end{displaymath} (5)

¤û¹y¦Ò¼{ªº¬O

\begin{displaymath}
f_n(x)=\int^x_0(1-x^2)^{\frac{n}{2}}dx
\end{displaymath}

·í n ¬°°¸¼Æ®É¡Afn(x) ³£¥i¨D±o¡]¬°¦h¶µ¦¡¡^¡A¥L¦A±q³o¨Ç¦h¶µ¦¡¨t¼Æ¶¡¡]°ò¥»¤W¬O Pascal ¤T¨¤§Î¡^ªºÃö«Y¡A¥Î¤º´¡ªk±Àºt¥X f1(x) ªº«Y¼Æ¡A¦Ó±o
f1(x) = $\displaystyle \int^x_0(1-x^2)^{n/2}dx$  
  = $\displaystyle x-\frac{\frac{1}{2}x^3}{3}-\frac{\frac{1}{8}x^5}{5}-\frac{\frac{1}{16}x^7}{7}-\frac{\frac{5}{128}x^9}{9} - \cdots$ (6)

¦]¦¹
\begin{displaymath}
(1-x^2)^{\frac{1}{2}} = 1 - \frac{1}{2} x^2 - \frac{1}{8}x^4 - \frac{1}{16}x^6 - \frac{5}{128}x^8 \cdots
\end{displaymath} (7)

¤û¹y»P Wallis ³Ì¤jªº¤£¦P³B¬O¡A¥L§â¿n¤Àªº¤W­­¥Ñ©T©wªº¼ÆÅܦ¨ÅܼơA¦]¦¹±o¨ì¾­¯Å¼Æªºªí¥Ü¡C¦b(6)¤¤¡AÅý x=1 ´N±o
\begin{displaymath}
\frac{\pi}{4} = 1 - \frac{1}{6} -\frac{1}{40} - \frac{1}{112} - \frac{5}{1152} - \cdots
\end{displaymath} (8)

¦Ó¦b(7)¤¤¡A¥Lª`·N¨ì (-x2)k ªº«Y¼Æ¥iªí¦¨¬°

\begin{displaymath}
\frac{1}{k!} \frac{1}{2}(\frac{1}{2}-1)(\frac{1}{2}-2)
\cdots (\frac{1}{2}-k+1)
\end{displaymath}

¥Ñ¦¹±À¼s´N±o¤@¯ëªº¤G¶µ¾­¯Å¼Æ®i¶}¦¡(2)¡B(3)¡C

·íµM¡A¤û¹yª¾¹D¤º´¡±Àºt¨Ã¤£¬OÃÒ©ú¡F¬°¤F¸ÕÅç¥Lªºµ²ªG¡A ¥L§â $(1-x^2)^{\frac{1}{2}}$ ªº¾­¯Å¼Æ®i¶}¦Û­¼¤@¦¸¡Aµ²ªGµo²{°£¤F 1-x2 ¥~¡A¾l¶µ³£®ø¥¢¤F¡C ¥L¤S§â $(1-x^2)^{\frac{1}{2}}$ ¬Ý¦¨¬O¤@­Ó¼Æ 1-x2 ªº¶}¤è¡A ¦Ó¥Î¥­±`ªº¶}¤èªk§@§Î¦¡¤Wªº¶}¤è¡A¤]±o¨ì¦P¼Ëªº¾­¯Å¼Æ®i¶}¦¡¡A³oµ¹¤F¤û¹y«Ü¤jªº«H¤ß¡G ¥L¤£¦ý½T©w¤G¶µ®i¶}¦¡¬O¹ïªº¡A¦Ó¥B¹ï¾­¯Å¼Æ¥Î¤@¯ë¥N¼Æªº¤èªk¨Ó¹Bºâ¤]´N¤£¦A¿ðºÃ¤F¡C

(1)¦¡ªº­nºò³B¬O¤À¥À h ³Q¬ù±¼¤F¡C³o¦b¦h¶µ¦¡¬O¿ì±o¨ìªº¡F­Y§â¨ç¼Æªí¦¨¾­¯Å¼Æ®É¤]¿ì¨ì¤F¡C²{¥Nªº·L¿n¤À½Ò¥»¡A¦b³B²z¤T¨¤¨ç¼Æªº·L¿n¤À®É¡A³£±q¥¿©¶¨ç¼Æ¥Xµo¡A¦Ó¥BµL½×¬O¥ý°µ¥¦ªº·L¤À©Î¥ý°µ¿n¤À¡AÁ`§K¤£¤F­n³B²z $\lim_{h \rightarrow 0} \sin \frac{h}{h}$¡C ³oùتº¤À¥À¬O¬ù¤£±¼ªº¡A©Ò¥H³o­Ó·¥­­­Èªº³B²z´N¬O²{¥N¤T¨¤¾Ç·L¿n¤Àªº­«ÂI©Ò¦b¡C §Ú­Ì¤£ª¾¹D¤û¹y·|¤£·|³B²z¥¦¡A¦ý§Ú­Ìª¾¹D¤û¹y¬ã¨s¤T¨¤¨ç¼Æªº·L¿n¤À¬O±q $\sin^{-1}x$ ¾­¯Å¼Æ®i¶}¤J¤âªº¡C



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¦p¹Ï¤@¡AQ ¬°³æ¦ì¶ê¤Wªº¤@ÂI¡AQP ¬°««½u¡APªº§¤¼Ð¬° x ¡A«h $\angle QOP$ ¬° cos-1x¡A¦]¦¹ $\theta = \angle QOR = sin^{-1}x$¡A ¦Ó®°§Î QOR ªº­±¿n¥¿¦n¬O $\frac{1}{2} \theta$¡C ¥t¤@¤è­±¡A®°§Î QOR ªº­±¿n¥¿¦n¬O¦±½u RQ¤Uªº­±¿n´î¥h¤T¨¤§Î OPQ ªº­±¿n¡A ¦]¦¹

\begin{displaymath}
\sin^{-1}x = 2(\int^x_0 \sqrt{1-x^2}dx-\frac{1}{2}x\sqrt{1-x^2})
\end{displaymath} (9)

³o®É«á¡A¤û¹y¥Î¤W¤F $(1-x^2)^{\frac{1}{2}}$ ªº¾­¯Å¼Æ¡A ¦A¸g³v¶µ¿n¤À¤Î¨â¯Å¼Æªº¦X¨Ö¡A¥L´N±o¨ì
\begin{displaymath}
\sin^{-1}x = \sum^{\infty}_{n=1} a_{2n-1} x^{2n-1}
\end{displaymath} (10)


\begin{displaymath}
a_{2n-1}=\frac{1^2 \cdot 3^2 \cdots (2n-3)^2}{(2n-1)!} , n \leq 2; a_1=1
\end{displaymath} (11)

°²³] $x=\sin \theta$ ªº¾­¯Å¼Æ¬°
\begin{displaymath}
x= \sin \theta =\sum^{\infty}_{n=0} b_m \theta^m
\end{displaymath} (12)

±N¥¦¥N¤J(10)¦¡¡A±o
\begin{displaymath}
\theta = \sin^{-1}x = \sum^{\infty}_{n=1} a_{2n-1} (\sum^{\infty}_{m=0}b_m\theta^m)^{2n-1}
\end{displaymath} (13)

¤ñ¸û¨âÃä £c ¦¸¤èªº«Y¼Æ¡A´N±o
\begin{displaymath}
b_{2m} =0 , b_{2m-1}=(-1)^{m-1}\frac{1}{(2m-1)!}
\end{displaymath} (14)

¥ç§Y
\begin{displaymath}
\sin \theta = \sum^{\infty}_{m=1} \frac{(-1)^{m-1}\theta^{2m-1}}{(2m-1)!}
\end{displaymath} (15)

¥ÎÃþ¦üªº¤èªk¥i±o
\begin{displaymath}
\cos \theta = \sum^{\infty}_{m=1} \frac{(-1)^{m-1}\theta^{2m}}{(2m)!}
\end{displaymath} (16)

¥Ñ(15)¡B(16)¨â¦¡¡A¥Î³v¶µ·L¤À¤Î¿n¤Àªº¿ìªk¡A¤û¹y±o¨ì¥¿¡B¾l©¶¨â¨ç¼Æªº·L¤À»P¿n¤À¡A

$\tan^{-1}x$ ªº¾­¯Å¼Æ¤]¥i¥Î¨Ó­pºâ¶ê©P²v¡A¬O·L¿n¤Àªº­«­n½ÒÃD¤§¤@¡A²{¥Nªº·L¿n¤À¬O

\begin{displaymath}
\tan^{-1}x = \int^x_0 \frac{dx}{1+x^2}
\end{displaymath} (17)

¥Xµo¡A±N¤G¶µ¦¡ (1+x2)-1 ¥H¡]´X¦ó¡^¾­¯Å¼Æ®i¶}¡A¦A³v¶µ¿n¤À¦Ó±o
\begin{displaymath}
\tan^{-1}x = x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7} + \cdots
\end{displaymath} (18)

³o¼Ë°µªkªº¥ý¨M±ø¥ó¬Oª¾¹D $\tan^{-1}x$ ªº·L¤À´N¬O $\frac{1}{1+x^2}$¡F¦ý¥¿¦n¬Û¤Ï¡D¤û¹y¦b³B²z©Ò¿× ¡uAgnesi ªº¤k§Å¡v µù1 $y = \frac{1}{1+x^2}$ ³oºØ¦±½u¤Uªº­±¿n®É¡F ¥ýª¾¹D $\frac{1}{1+x^2}$ ªº¿n¤À´N¬O $\tan^{-1}x$¡A¦]¦¹±o¨ì tan-1x ªº¾­¯Å¼Æ(18)¡C



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¦p¹Ï¤G¡A³] OP¡BAQ ¬°¤@¶êª½®| OA ¨âºÝªº¤Á½u¡A³] PQ ««ª½©ó¤Á½u¡A ¦Ó OQ ¥æ¶ê©ó B¡ABC ¥­¦æ©ó¤Á½u¦Ó¥æ PQ ©ó C¡C­Y P ¬°°ÊÂI¡A «h C ªº­y¸ñ´N¬O©Ò­nªº¦±½u¡C­Y§â OP, QA ¤À§O·í¦¨ x »P y ¶b¡A ¥O OA ªºªø«×¬°1¡AP ÂIªº§¤¼Ð¬° x¡A«h C ÂIªº°ª«×¥¿¬O $\frac{1}{1+x^2}$¡A ³o¥¿¬O¦±½uªº¨ç¼Æ¡C¦Ó¤û¹y­n­pºâªº¥¿¬O¥k¦¡ªº¿n¤À¡C¤û¹y§âÅܼƴ«¦¨ t=OB¡C ¥Ñ OB:OQ = PC:PQ¡A¥i±o

\begin{displaymath}
t = OQ \cdot PC = \sqrt{1+x^2} \cdot \frac{1}{1+x^2} = \frac{1}{\sqrt{1+x^2}}
\end{displaymath}

©Î

\begin{displaymath}
x = \frac{\sqrt{1-t^2}}{t}
\end{displaymath}



¹Ï¤T

¦]¦¹

$\displaystyle \int^x_0\frac{dx}{1+x^2}$ = $\displaystyle \int^t_1(\frac{-t^2}{\sqrt{1-t^2}}
- \sqrt{1-t^2})dt$  
  = $\displaystyle \int^t_1 td\sqrt{1-t^2}-\int^t_1\sqrt{1-t^2}dt$  
  = $\displaystyle t \sqrt{1-t^2} \vert^t_1 - s \int^t_1\sqrt{1-t^2}dt$  
  = $\displaystyle t \sqrt{1-t^2} + 2 \int^1_t \sqrt{1-t^2}dt$ (19)

³o­Ó®É­Ô¡A½Ð¬Ý¹Ï¤T¡C³]¶ê¥b®| OD =1 , OE = t¡A «h $OD = \sqrt{1-t^2}$ ¡C¦]¦¹¤W¦¡ªº²Ä¤@¶µ¬° $\triangle ODE$ ­±¿nªº¨â­¿¡A¦Ó²Ä¤G¶µ«h¬°¦±½u DF ¤U­±¿nªº¨â­¿¡C©Ò¥H¥Ñ¤W¦¡¥i±o
$\displaystyle \int^x_0\frac{dx}{1+x^2}$ = $\displaystyle 2 ( \mbox{{\fontfamily{cwM5}\fontseries{m}\selectfont \char 109}\...
...0pt plus0.2pt minus0.1pt{\fontfamily{cwM2}\fontseries{m}\selectfont \char 9}} )$  
  = $\displaystyle \angle DOE = \tan^{-1}\frac{\sqrt{1-t^2}}{t} = \tan^{-1}x$ (20)

¦³¤F¤G¶µ¾­¯Å¼Æ®i¶}¦¡¡A¤û¹y±o¥H¦b·L¿n¤À¦³©Ò¬ð¯}¡A¤@¯ë¾­¯Å¼Æ¤]´NÅܦ¨¤F¤û¹y·L¿n¤Àªº¥D­n¤u¨ã¡C·L¿n¤À¦b¥L¤â¤¤¦¨¤F¸àÄÀ¦ÛµM²{¶H³ÌµR§Qªº¤u¨ã¡C ¦b²{¥Nªº·L¿n¤À½Ò¥»¤¤¡A¾­¯Å¼Æ¥u¬O¨ä¤¤ªº¤@³¹¡A¦Ó¥B¬O¦b¾ã¥»®Ñªº«á¥b¡A¦Ó¤G¶µ®i¶}¦¡³»¦h¬O¨ä¤¤ªº¤@¤p¸`¡A¬Æ¦ÜÁÙ¥i¯à¤£¨£Âܼv¡A¤û¹y¦A¥Í¡A·í·P¹Ä¤£¤v¡C

 
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