Nonlinear equation containing parametric integral

Question

I have the following equation: $$I(k,x)=\int_0^xJ_k(\tau^k)d\tau=\alpha$$ where $\alpha$ is a given constant $\alpha\in \mathbb{R}$ and $k$ integer with $k\gt 0$. $J_k(x)$ is the Bessel function of first kind. My question is: is it possible to solve the equation for every $k$ or there is only a couple $(k_0,x_0)$, solving the $I(k,x)=\alpha$?

Thanks in advance for suggestions

Answer 1

For $\alpha \in \mathbb{R}$ this can't be solved for every $k$. A simple counter-example shows this.

Consider $\alpha > 1 - J_{0}(j_{1,1})$ and $k = 1$.

Then: $$ \int_{0}^{x} J_{1}(\tau) d\tau = 1 - J_{0}(x) \leq 1 - J_{0}(j_{1,1}) < \alpha.$$ This integral is always smaller than $\alpha$, regardless of $x$.