Question

如何用matlab求导数

Answer 1

Γ(α)=∫_0^∞〖x^(α-1) e^(-x) dx〗

Γ(1/2)=∫_0^∞〖x^(-1/2) e^(-x) dx〗

=∫_0^∞〖e^(-x) d(2x^(1/2))〗

=∫_0^∞〖e^(-u^2 ) d(2u) 〗 (令u=x^(1/2) )

=∫_0^∞〖2·e^(-u^2 ) du〗

=∫_(-∞)^∞〖e^(-u^2 ) du〗

Γ(1/2)·Γ(1/2)

=∫_(-∞)^∞〖e^(-u^2 ) du〗·∫_(-∞)^∞〖e^(-v^2 ) dv〗

=∫_(-∞)^∞∫_(-∞)^∞〖e^(-u^2-v^2 ) dudv〗

=∫_0^2π  ∫_0^∞〖e^(-r^2 ) rdrdθ〗
=∫_0^∞〖e^(-r^2 ) rdr〗 ∫_0^2π  dθ

=-1/2 e^(-r^2 ) |(0，∞) ·∫_0^2π dθ

=1/2·2π

=π

Γ(1/2)=√π