# Bessel Functions Cheat Sheet

• MHB
• Ackbach

#### Ackbach

Gold Member
MHB
This is a helpful document I got from one of my DE's teachers in graduate school, and I've toted it around with me. I will type it up here, as well as attach a pdf you can download.

Bessel Functions​

$$J_{\nu}(x)=\sum_{m=0}^{\infty}\frac{(-1)^{m}x^{\nu+2m}}{2^{\nu+2m} \, m! \,\Gamma(\nu+m+1)}$$
is a Bessel function of the first kind of order $\nu$. The general solution of $x^2 \, y''+x \, y'+(x^2-\nu^2) \, y=0$ is $y=c_1 \, J_{\nu}(x)+c_2 \, J_{-\nu}(x)$. If $\nu=n$ is an integer, the general solution is $y=c_1 \, J_n(x)+c_2 \, Y_n(x)$ where $Y_n(x)$ is the Bessel function of the second kind of order $n$. Here, $Y_n(x)$ equals $\frac{2}{\pi} \, \ln\left(\frac{x}{s}\right)$ plus a power series.

The solutions of $x^2 \, y''+x \, y'+(-x^2-\nu^2) \, y=0$ are expressible in terms of modified Bessel functions of the first/second kind of order $\nu$, namely $I_{\nu}(x)$ and $K_{\nu}(x)$.

The graphs:

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You can use these graphs sometimes to work out initial conditions, particularly if any of them are zero.

Equations Solvable in Terms of Bessel Functions​

If $(1-a)^2\ge 4c$ and if neither $d$, $p$ nor $q$ is zero, then, except in the obvious special case when it reduces to the Cauchy-Euler equation $(x^2 y''+axy'+cy=0)$, the differential equation
$$x^2y''+x(a+2bx^p)y'+[c+dx^{2q}+b(a+p-1)x^p+b^2x^{2p}]y=0$$
has as general solution
$$y=x^{\alpha} \, e^{-\beta x^p} [C_1 \, J_{\nu}(\varepsilon x^q)+C_2 Y_{\nu}(\varepsilon x^q)]$$
where
$$\alpha=\frac{1-a}{2}, \qquad \beta=\frac{b}{p},\qquad \varepsilon=\frac{\sqrt{|d|}}{q},\qquad \nu= \frac{\sqrt{(1-a)^2-4c}}{2q}.$$
If $d<0$, then $J_{\nu}$ and $Y_{\nu}$ are to be replaced by $I_{\nu}$ and $K_{\nu}$, respectively. If $\nu$ is not an integer, then $Y_{\nu}$ and $K_{\nu}$ can be replaced by $J_{-\nu}$ and $I_{-\nu}$ if desired.

The following file is a pdf of the above.

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#### Attachments

• J0Plot.pdf
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• J1Plot.pdf
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• I0Plot.pdf
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• I1Plot.pdf
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• K1Plot.pdf
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• Y1Plot.pdf
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• Bessel Function Cheat Sheet.pdf
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• Bessel Function Cheat Sheet.pdf
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That will be helpful to many students. Thanks for taking the time to type it up and for sharing.

• Ackbach