How to integrate product of Bessel and exponential fucntion

Question

I have obtained the following solution for inhomogeneous Helmholtz equation

\begin{align*} W(u) = \dfrac{i}{2 \lambda} e^{i \lambda u} \int_{0}^u J_{n}(\lambda u^{'})e^{-i \lambda u^{'}} du^{'} \end{align*} Could someone please help me on how to integrate this product of two functions MATHEMATICA? Thank you.

Answer 1

Substitution z->\[Lambda] u,\[Xi]->\[Lambda] u' helps to reduce the integral to W[u]=int[\[Lambda] u,n]/\[Lambda]^2

with

int[z_?NumericQ, n_?NumericQ] :=Block[{\[Xi]},I/ 2  Exp[I z] NIntegrate[BesselJ[n, \[Xi]] Exp[-I\[Xi]], {\[Xi], 0, z}]]

which only depends on two parameters .

Answer 2

One gets nice analytical results by specifying the order of the Bessel function. For instance

Integrate[Exp[-I x] BesselJ[0, x], {x, 0, u}, Assumptions -> u > 0]

$$e^{-i u} u \left(J_0(u)+i J_1(u)\right).$$

By following this systematically one obtains the following

$$\int_0^u J_n(x) e^{i x}dx =-n + e^{-i u} \frac{P_{n-2}(u)}{u^{n-3}} J_0(u) + e^{-i u} \frac{Q_{n-1}(u)}{u^{n-2}} J_1(u),$$

where $P_n(u)$ and $Q_n(u)$ are certain $n$-order polynomials of $u$.