Zurück zu Bestimmte Integrale
In der Formel ∫ 0 ∞ [ W ( 1 x 2 ) ] α d x = α ⋅ 2 α − 1 / 2 ⋅ Γ ( α − 1 2 ) {\displaystyle \int _{0}^{\infty }\left[W\left({\frac {1}{x^{2}}}\right)\right]^{\alpha }dx=\alpha \cdot 2^{\alpha -1/2}\cdot \Gamma \left(\alpha -{\frac {1}{2}}\right)} setze α = 1 {\displaystyle \alpha =1} , dann ist ∫ 0 ∞ W ( 1 x 2 ) d x = 2 ⋅ Γ ( 1 2 ) = 2 π {\displaystyle \int _{0}^{\infty }W\left({\frac {1}{x^{2}}}\right)\,dx={\sqrt {2}}\cdot \Gamma \left({\frac {1}{2}}\right)={\sqrt {2\pi }}} .
∫ 0 ∞ W ( 1 x 2 ) d x {\displaystyle \int _{0}^{\infty }W\left({\frac {1}{x^{2}}}\right)dx} ist nach Substitution x ↦ x − 1 / 2 {\displaystyle x\mapsto x^{-1/2}} gleich 1 2 ∫ 0 ∞ W ( x ) x x d x {\displaystyle {\frac {1}{2}}\,\int _{0}^{\infty }{\frac {W(x)}{x\,{\sqrt {x}}}}\,dx} .
Die Funktion f ( x ) = [ W ( 1 x 2 ) ] α {\displaystyle f(x)=\left[W\left({\frac {1}{x^{2}}}\right)\right]^{\alpha }} besitzt die Umkehrfunktion g ( x ) = x − 1 2 α ⋅ e − 1 / 2 ⋅ x 1 / α {\displaystyle g(x)=x^{-{\frac {1}{2\alpha }}}\cdot e^{-1/2\cdot x^{1/\alpha }}} . Nun ist ∫ 0 ∞ f ( x ) d x = ∫ 0 ∞ g ( x ) d x = ∫ 0 ∞ g ( x α ) α x α − 1 d x {\displaystyle \int _{0}^{\infty }f(x)\,dx=\int _{0}^{\infty }g(x)\,dx=\int _{0}^{\infty }g(x^{\alpha })\,\alpha \,x^{\alpha -1}\,dx} = α ∫ 0 ∞ x − 1 / 2 + α − 1 e − 1 / 2 ⋅ x d x = α ⋅ 2 α − 1 / 2 ⋅ Γ ( α − 1 2 ) {\displaystyle =\alpha \int _{0}^{\infty }x^{-1/2+\alpha -1}\,e^{-1/2\cdot x}\,dx=\alpha \cdot 2^{\alpha -1/2}\cdot \Gamma \left(\alpha -{\frac {1}{2}}\right)} .
![{\displaystyle \int _{0}^{\infty }\left[W\left({\frac {1}{x^{2}}}\right)\right]^{\alpha }dx=\alpha \cdot 2^{\alpha -1/2}\cdot \Gamma \left(\alpha -{\frac {1}{2}}\right)\qquad {\text{Re}}(\alpha )>{\frac {1}{2}}}](./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2cf44400cffd7aa5afe8e9b22ecc0730fd1bd8b.svg)
![{\displaystyle \int _{0}^{\infty }\left[W\left({\frac {1}{x^{2}}}\right)\right]^{\alpha }dx=\alpha \cdot 2^{\alpha -1/2}\cdot \Gamma \left(\alpha -{\frac {1}{2}}\right)}](./_assets_/eb734a37dd21ce173a46342d1cc64c92/627fd6a040c27c73e839d69ed2eae60709dded42.svg)
![{\displaystyle f(x)=\left[W\left({\frac {1}{x^{2}}}\right)\right]^{\alpha }}](./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c4d6852c68d848f9c730b4e5dc491e176f874a3.svg)
