フェルマーの最終定理の証明 (639ﾚｽ)
上下前次1-新
抽出解除 必死ﾁｪｯｶｰ(本家) (べ) 自ID ﾚｽ栞 あぼーん

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535: 07/30(水)11:29 ID:O8+PnNqB(1/3)調 AAS
f(t)=t (-π<t?π)
b_n=∫_(-π)^π?f(t)sin(nt) dt=∫_(-π)^π?tsin(nt) dt
=-∫_(-π)^π??t(cos(nt)/n)^' ? dt

=-[t/n cos(nt)]_(-π)^π+1/n ∫_(-π)^π?cos(nt) dt
=-(π/n cos(nπ)-(-π)/n cos(-nπ))+1/n [1/n sin(nt)]_(-π)^π
=-(π/n cos(nπ)+π/n cos(nπ))=2π (-cos(nπ))/n=2π (-1)^(n+1)/n
f(t)=2?_(n=1)^∞?(-1)^(n+1)/n sin(nt)
=2(sin(t)-1/2 sin(2t)+1/3 sin(3t)-1/4 sin(4t)+1/5 sin(5t)-?)
f(π/2)=π/2=sin(π/2)+1/3 sin(3π/2)+1/5 sin(5π/2)+1/7 sin(7π/2)+?
=1-1/3+1/5-1/7+?=?_(n=1)^∞?(-1)^(n+1)/(2n-1)
∴π=2?_(n=1)^∞?(-1)^(n+1)/(2n-1)

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536: 07/30(水)11:30 ID:O8+PnNqB(2/3)調 AAS
∫_0^∞?(sin(x))/x dx
∂/∂s (e^(-sx) (sin(x))/x)=-xe^(-sx) (sin(x))/x=-e^(-sx) sin(x)

F(s)=∫_0^∞??e^(-sx) (sin(x))/x? dx (s?0)

dF(s)/ds=d/ds ∫_0^∞??e^(-sx) sin?(x)/x? dx
=∫_0^∞??∂/ds e^(-sx) sin?(x)/x? dx
=∫_0^∞??-xe^(-sx) sin?(x)/x? dx=-∫_0^∞??e^(-sx) sin?(x) ? dx
=-∫_0^∞??-1/s (e^(-sx) )^' sin(x)? dx
=∫_0^∞??1/s (e^(-sx) )^' sin(x)? dx
=[1/s e^(-sx) sin(x)]_0^∞-1/s ∫_0^∞??e^(-sx) cos(x)? dx
=0-1/s ∫_0^∞??e^(-sx) cos(x)? dx=-1/s ∫_0^∞???-1/s (e^(-sx) )?^' cos(x)? dx
=1/s^2 ∫_0^∞??(e^(-sx) )^' cos(x)? dx
=[1/s^2 e^(-sx) cos(x)]_0^∞-1/s^2 ∫_0^∞??-e^(-sx) sin(x)? dx
=-1/s^2 +1/s^2 ∫_0^∞??e^(-sx) sin(x)? dx
=-1/s^2 -1/s^2 dF(s)/ds (dF(s)/ds=-∫_0^∞??e^(-sx) sin?(x) ? dx)

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537: 07/30(水)11:31 ID:O8+PnNqB(3/3)調 AAS
dF(s)/ds+1/s^2 dF(s)/ds=-1/s^2
(1+1/s^2 ) dF(s)/ds=((s^2+1)/s^2 ) dF(s)/ds=-1/s^2
dF(s)/ds=-1/(s^2+1)
F(s)=-∫?1/(s^2+1) ds s=tan(θ) ds=1/(?cos?^2 (θ) ) dθ
-∫?1/(s^2+1) ds=-∫??1/(?tan?^2 (θ)+1)?1/(?cos?^2 (θ) )? dθ=-θ=-arctan(s)+C

F(s)=-arctan(s)+C
F(s)=∫_0^∞??e^(-sx) sin?(x)/x? dx (s?0)
F(∞)=∫_0^∞?0 dx=0=-arctan(∞)+C
C=arctan(∞)=π/2
F(s)=-arctan(s)+π/2

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