# Tag Info

## Hot answers tagged bessel-functions

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### Integral with 4 Bessel functions and an exponential

Let's consider the second integral, which can be written in the following form: $$I(p, q, i, j, k, l; a, b, c, d) := \int_0^\infty dt\, \exp(-p t^2) t^q j_i(a t) j_j(b t) j_k(c t) j_l(d t)$$ where ...
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### Integral involving Bessel function

The integral can be expressed in terms of Bessel and Struve functions, $$I(\xi)=\int_0^{\infty} dq\, \frac{J_0(q \xi)}{q+1}=\tfrac{1}{2} \pi \bigl(\pmb{H}_0(\xi)-Y_0(\xi)\bigr).$$ The small-$\xi$ ...
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### Spherical Bessel functions. Sum of squares

From the definition 10.47.10, it follows that $$\mathsf{j}_{n}^{2}(z)+\mathsf{y}_{n}^{2}(z) = h_n^{(1)}(z)\cdot h_n^{(2)}(z).$$ So, by the expansions 10.49.6 and 10.49.7, \begin{split} \mathsf{j}_{n}^{...
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### A hypergeometric identity related to Bessel functions

In "Higher Trancendental Functions", Vol. 1, by A. Erdelyi (ed.), on page 86, equation (4) says $$_2F_0(α,β;z)\ _2F_0(α,β;-z) = \, _4F_1(α,β,\frac{1}{2}(α+β),\frac{1}{2}(α+β+1);α+β;4z^2),$$ from ...
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### Closed form for $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$

as requested by the OP in the comment section: $$\int_0^T e^{-x}I_n(x)\frac{1}{x}\,dx=\frac{1}{n}+\frac{1}{n T^{n-1}}e^{-T}\left[a_n(T)I_0(T)+b_n(T)I_1(T)\right]$$ the functions $a_n$ and $b_n$ are ...
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### Is there any integer $n$ such that the bessel function J_n(1)=0?

According to Uniform Upper and Lower Bounds on the Zeros of Bessel Functions of the First Kind, p 2: $$j_{v,k} > v + \frac23 |a_{k-1}|^{3/2}$$ where $j_{v,k}$ is the $k$-th positive zero of $J_v$...
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### Proof of an identity involving $\int \exp(-|x-s|)dx$ over an even sphere

Following on from the answer of Sam Dolan, I generalized my conjecture to the following, which I prove as Theorem 4 in my paper The magnitude of odd balls via Hankel determinants of reverse Bessel ...
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### A Bessel function integral identity involving $\int_0^\pi \frac{K_{j-1/2}(w)}{w^{j-1/2}}\sin^{2p-1}(\theta)\, d\theta$

I have proved this Bessel identity in my paper The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials. It is Theorem 4 which is proved in Section 3.
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### Is there any integer $n$ such that the bessel function J_n(1)=0?
The Bessel functions have the series expansion $$J_n(z) = \sum_{k=0}^\infty \dfrac{(-1)^k (z/2)^{n+2k}}{k! (k+n)!}$$ If $0 < z \le 2 \sqrt{1+n}$ this is a convergent alternating series with ...
Mathematica can evaluate the integral$^\ast$ I_{n,l}=\begin{align} \int_{0}^{\infty } e^{-\frac{r^2}{2B}} r^{l-n} L_n^{l-n}\left(\frac{r^2}{C}\right) I_{l-n}\left(\rho r \right) r dr \end{...