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#46
22 ÁԶعÒ¹ 2006, 20:55
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¼ÁʧÊѹԴ¹Ö§¤ÃѺÇèÒàÃҨзÓ2¢é͹Õéä´éËÃ×ÍäÁèËÒ¡äÁèãªéá¤Å¤ÙÅÑʪèÇÂàÅÂẺÇèÒÁ.»ÅÒÂÅéǹæ
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$$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x-b\sin x}{a\sin x+b\cos x}dx=\ln\left(\frac{a}{b}\right)$$ BUT $$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x+b\sin x}{a\sin x+b\cos x}dx=\frac{\pi ab}{a^{2}+b^{2}}+\frac{a^{2}-b^{2}}{a^{2}+b^{2}}\ln\left(\frac{a}{b}\right)$$
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#47
23 ÁԶعÒ¹ 2006, 19:49
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´Ù¨Ò¡¤ÓµÍº¢Í§·Ñé§ 2 ¢éÍáÅéÇ ¤Ô´ÇèÒÇÔ¸ÕÁ.»ÅÒ ÍÂèÒ§à´ÕÂÇ ¤§àÍÒäÁèÍÂÙèËÃÍ¡¤ÃѺ àÇé¹àÊÕÂáµèÇèÒâçàÃÕ¹¹Ñé¹æ ¨ÐÊ͹à¡Ô¹ËÅÑ¡ÊÙµÃ
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ
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#49
23 ÁԶعÒ¹ 2006, 21:40
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µÑÇÍÂèÒ§ÅͧÈÖ¡ÉÒ¨Ò¡µÑÇÍÂèÒ§¹Õé¤ÃѺ
Evaluate $$\displaystyle{ \frac{9!}{15!}+\frac{12!}{18!}+\frac{15!}{21!}+\frac{18!}{24!}+\dots } $$ $\frac{3541 - 231\sqrt{3}\pi - 2079 \ln 3}{55440}$ Solution
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#50
24 ÁԶعÒ¹ 2006, 01:45
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ÍéÍ ä´éÁÕ¡ÒäØÂàÃ×èͧ¹Õé¡Ñ¹Â¡ãËèä»áÃéǹÕè¹Ò ʧÊѪèǧ¹Ñé¹äÁèä´éà¢éÒºÍÃì´¤ÃѺ à´ëÇ¢Íä»·Ó¤ÇÒÁà¢éÒ㨡è͹¹Ð¤ÃѺ áÃéǶéÒÁÕ»ÑËÒ䧨ÐÁÒ¶ÒÁãËÁè¤ÃѺ
¢Íº¤Ø³ ¤Ø³ Mastermander ´éǤÃѺ¼Á »Å. äÁè·ÃÒºÇèÒ ¤Ø§ Mastermander ¹Õè àÃÕ¹ªÑé¹ÍÒÃÑÂËÃͤÃѺ àËç¹ power áçÁÒ¡æ ÍÔÍÔ ¶ÅèÁºÍÃì´ Vcharkarn ¡ÃШÒ ·Õè¹Õè¡çŧ¡ÃзÙé Marathon ¡Ñ¹ãËè (¼ÁäÁè¤èÍÂà¢éÒ Marathon ËÃÍ¡¤ÃѺ àÂÕèÂÁàÂÕ¹áÇêºæ ¨ÐµÍº»ÑËÒÂèÍÂæ«ÐÁÒ¡¡ÇèÒ äÁè¤èÍÂÁÕàÇÅÒ«Ñ¡à·èÒääÃѺ T_T )
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#51
24 ÁԶعÒ¹ 2006, 16:18
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¼ÁàÃÕ¹ÍÂÙè Á.6 ¤ÃѺ
Êèǹ¹éͧ Timestopper à¤éÒàÃÕ¹ÍÂÙè Á.3 ¤ÃѺ (ÍѨ©ÃÔÂеÑǨÃÔ§)
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒã¨
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#52
25 ÁԶعÒ¹ 2006, 16:48
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àÅè¹·Ó¼ÁÅÍÂËËÑǪ¹¡ÓᾧàŤÃѺ ¡è͹ª¹->  ,ËÅѧª¹-> 
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$$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x-b\sin x}{a\sin x+b\cos x}dx=\ln\left(\frac{a}{b}\right)$$ BUT $$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x+b\sin x}{a\sin x+b\cos x}dx=\frac{\pi ab}{a^{2}+b^{2}}+\frac{a^{2}-b^{2}}{a^{2}+b^{2}}\ln\left(\frac{a}{b}\right)$$
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#53
26 ÁԶعÒ¹ 2006, 16:08
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Evaluate
$$ \prod_{n=1}^\infty n^{1/{n^2}} $$ how to solve them ?
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒã¨
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#55
26 ÁԶعÒ¹ 2006, 21:38
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¢Íº¤Ø³¤ÃѺ
¨Ò¡¡Òá´à¤Ã×èͧ¨Ðä´é $$ e^{-Zeta'(2)}=2.55371 $$ «Ö觼Á§§¡Ñº Zeta'(2) ÇèÒÁѹ¤×ÍÍÐäÃ
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒã¨
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#56
27 ÁԶعÒ¹ 2006, 12:03
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zeta ã¹·Õè¹ÕéËÁÒ¶֧ Riemann's zeta function: $$ \zeta(x):= \sum_{n=1}^\infty \frac{1}{n^x} $$ àÁ×èÍ $x>1$
zeta' ¡ç¤×Í derivative ¢Í§ $\zeta(x)$ ¹Ñ蹤×Í $$\zeta'(x) = -\sum_{n=1}^\infty \frac{\ln n}{n^x} $$ ´Ñ§¹Ñé¹àÃÒ¨Ö§ä´éÇèÒ $$ \ln \prod_{n=1}^\infty n^{1/{n^2}} = \sum_{n=1}^\infty \frac{\ln n}{n^2} = -\zeta'(2) $$ ´Ñ§·Õèà¤Ã×èͧºÍ¡äÇé¤ÃѺ 
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#57
07 ¡Ã¡®Ò¤Á 2006, 00:03
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒã¨
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#58
09 ¡Ã¡®Ò¤Á 2006, 23:47
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PolyLog ¤×ÍÍÐääÃѺ
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒã¨
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#59
10 ¡Ã¡®Ò¤Á 2006, 12:14
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¹ÔÂÒÁ¢Í§ Mathematica PolyLog ´Ùä´é·Õè¹Õè¤ÃѺ
¨Ò¡ $$ PolyLog[\nu,z] := Li_{\nu}(z) = \sum_{k=1}^\infty \frac{z^k}{k^\nu} $$ ´Ñ§¹Ñé¹ $$ \frac{\partial}{\partial \nu} PolyLog[\nu,z] = - \sum_{k=1}^\infty \frac{z^k \ln k}{k^\nu} $$ àÃÒ¨Ö§ä´éÇèÒ $$ \sum_{k=1}^\infty \frac{\ln k}{2^k} = \left. - \frac{\partial}{\partial \nu} PolyLog[\nu,z] \right|_{ (\nu,z) = (0,\frac12) } $$ ÍÂèÒ§·Õè Mathematica ºÍ¡ ÊèǹÍѹËÅѧ¼ÁÇèÒ¾Í Mathematica à¨Í expression ÂÒ¡æà¢éÒ仺èÍÂæ ¡çªÑ¡¨ÐàÃÔèÁÁÑèÇàËÁ×͹¡Ñ¹¤ÃѺ
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#60
12 ¡Ã¡®Ò¤Á 2006, 20:58
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áÅéÇ͹ءÃÁ
$$ \sum_{n=1}^{\infty} \frac{\ln(n+1)}{2^n}$$ ÁդӵͺäËÁ¤ÃѺ
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒã¨
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