# Bessel function Calculator

## Calculates the Bessel functions of the first kind Jv(x) and second kind Yv(x), and their derivatives J'v(x) and Y'v(x).

 order v real number x complex number 6digit10digit14digit18digit22digit26digit30digit34digit38digit42digit46digit50digit
 $\normal Bessel\ functions\ of\\\ the\ 1st\ kind\ J_\nu(x)\ and\ 2nd\ kind\ Y_\nu(x)\\[10](1)\ x^2y''+xy'+(x^2-\nu^2)y=0\\\hspace{25} y=c_1J_\nu(x)+c_2Y_\nu(x)\\[10](2)\ J_\nu(x)={\large\sum_{\small k=0}^{\small\infty}\frac{(-1)^k}{k!\Gamma(k+\nu+1)}(\frac{x}{2})^{\nu+2k}}\\\hspace{30} Y_\nu(x)={\large\frac{J_\nu(x)cos(\nu\pi)-J_{-\nu}(x)}{sin(\nu\pi)}}\\[10](3)\ J'_\nu(x)=J_{\nu-1}(x)-{\large\frac{\nu}{x}}J_\nu(x)=-J_{\nu+1}(x)+{\large\frac{\nu}{x}}J_\nu(x)\\\hspace{25}Y'_\nu(x)=Y_{\nu-1}(x)-{\large\frac{\nu}{x}}Y_\nu(x)=-Y_{\nu+1}(x)+{\large\frac{\nu}{x}}Y_\nu(x)\\[10](4)\ {\large e^{\frac{x}{2}(t-{\large\frac{1}{t}})}}={\large\sum_{\small n=-\infty}^{\small\infty}}J_n(x)t^n,\hspace{20} n=integer\\$
Bessel function
 [1-10] /37 Disp-Num5103050100200
[1]  2018/11/05 00:59   Male / 20 years old level / High-school/ University/ Grad student / Very /
Purpose of use
nuclear reactor power density calculations
[2]  2018/06/07 09:54   Male / 20 years old level / High-school/ University/ Grad student / Very /
Purpose of use
dynamic stability calculations
[3]  2017/12/10 03:28   Male / 40 years old level / An engineer / Very /
Purpose of use
To calculate ground displacements due to seismic and dynamic loads
Comment/Request
This is awesome!! Thanks for the great help and please keep up the good work.
[4]  2017/04/19 05:07   Male / 40 years old level / High-school/ University/ Grad student / Very /
Purpose of use
For communication signal analysis
[5]  2017/03/01 23:52   Male / 20 years old level / High-school/ University/ Grad student / Not at All /
Purpose of use
To solve bandwidth calculation problems
[6]  2016/01/24 19:17   Male / 20 years old level / High-school/ University/ Grad student / Useful /
Purpose of use
To find out the Muller matrix elements of a modulator which involves sinusoidally varying retardance.
[7]  2015/09/18 04:15   Male / 60 years old level or over / A teacher / A researcher / Very /
Purpose of use
Fitting magnetic field for large solenoid for particle physics research. Needed to estimate J values to obtain reasonable starting values for fit parameters.
[8]  2015/09/16 14:51   Female / 30 years old level / A teacher / A researcher / Useful /
Purpose of use
for calculations to design resonant cavity
[9]  2015/05/26 02:37   Male / 50 years old level / A teacher / A researcher / Useful /
Purpose of use
Needed to calculate some values of J for research, but I was away from my main desktop machine.
[10]  2015/04/16 18:26   Male / 40 years old level / A teacher / A researcher / Very /
Purpose of use
I teach a class in numerical calculation - referred students to it. They compared website values with those computed by a upwards recursion.

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