# How to integrate product of Bessel and exponential fucntion

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.

• It has no known closed form: I/(2 λ) Exp[I λ u] Integrate[BesselJ[n, λ z] Exp[-I λ z], {z, 0, u}] - neither can Rubi solve it. Maybe you know values for λ and u and n in which case you could numerically integrate, otherwise I'm afraid you're out of luck. Jun 23, 2021 at 9:07
• Thanks for you answer. Yes, $\lambda$ and n are constants. Jun 23, 2021 at 9:21
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• Please add to your question the Mathematica code you have tried and the results you have obtained. Jun 23, 2021 at 11:47
• @Bharath simply saying they're constants doesn't help - you need to provide their values or this cannot be numerically integrated. Jun 23, 2021 at 12:07

## 2 Answers

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 .

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 $$u$$.

• Thanks for your answer. Its great to see some positive answer. Just a question. Any idea on how to chose polynomials $P_{n}(u)$ and $Q_{n}(u)$? @yarchik Jun 24, 2021 at 15:59
• @Bharath You can try to derive the recurrence relation between the polynomials using the recurrence relations for the Bessel functions. Try asking on math.stackexchange Jun 24, 2021 at 16:06