定理 Fresnel 积分定理

成立

$$ \bbox[5pt]{ \int_{0}^{\infty}{\sin(x^2)}{dx} = \int_{0}^{\infty}{\cos(x^2)}{dx} = \frac{\sqrt{2\pi}}{4} } $$

推广形式

$\forall n \in \mathbb{N}$

成立

$$ \bbox[5pt]{ \begin{aligned} \int_{0}^{\infty}{\sin(x^n)}{dx} & = \frac{1}{n}\Gamma\left(\frac{1}{n}\right)\sin{\frac{\pi}{2n}} \\ \int_{0}^{\infty}{\cos(x^n)}{dx} & = \frac{1}{n}\Gamma\left(\frac{1}{n}\right)\cos{\frac{\pi}{2n}} \end{aligned} } $$

证明