change of variables $ax-bx^{-1}$

hbghlyj

Theorem 2.1. [$\newcommand{\ift}{\int_{0}^{\infty}}\rm Cauchy$-$\rm Schl\ddot{o}milch$] Let $a, \, b >0$ and assume that $f$ is a continuous function for which the integrals are convergent. Then \begin{eqnarray*} \ift f \left( \left(ax - bx^{-1} \right)^{2} \right) \, dx & = & \frac{1}{a} \ift f(y^{2}) \, dy. \end{eqnarray*}

Proof

The change of variables $t = b/ax$ yields \begin{eqnarray*}I & = & \ift f \left( \left(ax - b/x \right)^{2} \right) \, dx \\ & = & \frac{b}{a} \ift f \left( \left(at - b/t \right)^{2} \right) \, t^{-2} \, dt. \end{eqnarray*} The average of these two representations, followed by the change of variables $u = ax - b/x$ completes the proof.

hbghlyj

Recall that 换元积分的函数φ需要为单调的
here $(0,\infty)\to(-\infty,\infty);ax-bx^{-1}$ is monotone and continuous

hbghlyj

Let $a, \, b >0$ and assume that $f$ is a continuous function for which the integrals are convergent and $f(x)=f(-x)$. Then \begin{eqnarray*} \ift f \left( ax - bx^{-1} \right) \, dx & = & \frac{1}{a} \ift f(y) \, dy. \end{eqnarray*}

hbghlyj

Glasser's Master Theorem

Solving $\int\limits_{-\infty}^\infty \frac{1}{x^8+1}dx$ through Glasser's Master Theorem