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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.

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    $\begingroup$ 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. $\endgroup$
    – flinty
    Jun 23 at 9:07
  • $\begingroup$ Thanks for you answer. Yes, $\lambda$ and n are constants. $\endgroup$
    – Bharath
    Jun 23 at 9:21
  • $\begingroup$ Welcome to Mathematica.SE! I hope you will become a regular contributor. To get started, 1) take the introductory tour now, 2) when you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge, 3) remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign, and 4) give help too, by answering questions in your areas of expertise. $\endgroup$
    – bbgodfrey
    Jun 23 at 11:46
  • $\begingroup$ Please add to your question the Mathematica code you have tried and the results you have obtained. $\endgroup$
    – bbgodfrey
    Jun 23 at 11:47
  • $\begingroup$ @Bharath simply saying they're constants doesn't help - you need to provide their values or this cannot be numerically integrated. $\endgroup$
    – flinty
    Jun 23 at 12:07
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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 .

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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 $u$.

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  • $\begingroup$ 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 $\endgroup$
    – Bharath
    Jun 24 at 15:59
  • $\begingroup$ @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 $\endgroup$
    – yarchik
    Jun 24 at 16:06

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