I am stuck with this integral: $$\int_0^\infty\frac{e^{-x}\ J_0(x)\ \sin\left(x\,\sqrt[3]{2}\right)}{x}dx,$$ where $J_0$ is the Bessel function of the first kind.

Is it possible to express this integral in a closed form (preferably, using elementary functions, Bessel functions, integers and basic constants)?


Hint: Use the formula $(79)$ from this MathWorld page: $$J_0(z)=\frac1\pi\int_0^\pi e^{i\,z\cos\theta}\,d\theta$$ and then change the order of integration.

Result: $$\int_0^\infty\frac{e^{-x}\ J_0(x)\ \sin\left(x\,\sqrt[3]2\right)}xdx=\arcsin\frac{\sqrt{2+\sqrt[3]4+\sqrt[3]{16}}-\sqrt{2+\sqrt[3]4-\sqrt[3]{16}}}2$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.