# Modified Bessel function Calculator

## Calculates the modified Bessel functions of the first kind Iv(x) and the second kind Kv(x), and their derivatives I'v(x) and K'v(x).

 order v real number x complex number 6digit10digit14digit18digit22digit26digit30digit34digit38digit42digit46digit50digit
 $\normal Modified\ Bessel\ functions\ of\\\ the\ 1st\ kind\ I_\nu(x)\ and\ 2nd\ kind\ K_\nu(x)\\[10](1)\ x^2y''+xy'-(x^2+\nu^2)y=0\\\hspace{25} y=c_1I_\nu(x)+c_2K_\nu(x)\\[10](2)\ I_\nu(x)={\large\sum_{\small k=0}^{\small\infty}\frac{1}{k!\Gamma(k+\nu+1)}(\frac{x}{2})^{\nu+2k}}\\\hspace{25} K_\nu(x)={\large\frac{\pi(I_{-\nu}(x)-I_\nu(x))}{2sin(\nu\pi)}}\\[10](3)\ I'_\nu(x)=I_{\nu-1}(x)-{\large\frac{\nu}{x}}I_\nu(x)=I_{\nu+1}(x)+{\large\frac{\nu}{x}}I_\nu(x)\\\hspace{25}K'_\nu(x)=-K_{\nu-1}(x)-{\large\frac{\nu}{x}}K_\nu(x)=-K_{\nu+1}(x)+{\large\frac{\nu}{x}}K_\nu(x)\\[10](4)\ {\large e^{\frac{x}{2}(t+{\large\frac{1}{t}})}}={\large\sum_{\small n=-\infty}^{\small\infty}}I_n(x)t^n,\hspace{20} n=integer\\$

Modified Bessel function
 [1-10] /15 Disp-Num5103050100200
[1]  2019/03/26 08:37   Male / 50 years old level / Others / Very /
Purpose of use
Modeling economic growth
Comment/Request
I wondering if the Modified Bessel function calculator is handling negative values for the order incorrectly. I am getting the same answer for v = 1, x = 2 as v = -1, x = 2. But I think these should not be the same.
[2]  2018/11/29 11:54   Male / 50 years old level / An engineer / Very /
Purpose of use
Sinusoidal modulation of exponential function around an operating point, for an RF peak detector.
Comment/Request
Saves time. No need to spend \$ on expensive Calculus handbook, nor programming to execute the numeric calculations. Thanks!
[3]  2017/03/27 01:52   Male / 30 years old level / An engineer / Very /
Purpose of use
Mean time to cycle slip calculation for first order bit tracking loop
[4]  2015/05/26 00:14   Male / 60 years old level or over / A teacher / A researcher / Very /
Purpose of use
Numerical solution of Poisson equation
[5]  2014/09/25 21:39   Male / 30 years old level / An engineer / Very /
Purpose of use
Calculation of bias current for a weak-inversion transistor in a crystal oscillator circuit
[6]  2014/09/06 01:36   Male / 60 years old level or over / A teacher / A researcher / Very /
Purpose of use
research in pure mathematics
[7]  2013/07/18 20:54   Male / 60 years old level or over / High-school/ University/ Grad student / Very /
Purpose of use
Modified Bessel functions for mathematical modelling for Honours School laboratory
[8]  2013/03/22 03:20   Male / 50 years old level / A teacher / A researcher / Very /
Purpose of use
Computation of Bessel and Kelvin functions with complex arguments
Comment/Request
It looks great!
[9]  2012/09/10 15:18   Male / 20 years old level / An office worker / A public employee / Very /
Purpose of use
Skin effect of a hollow conductor study
Comment/Request
excell 2010 can do same function too.
[10]  2011/10/27 14:42   Male / 60 years old level / A teacher / A researcher / Very /
Purpose of use
Evaluating Bessel functions of the first kind AND modified Bessel functions of the second kind. Certainly a quicker way than using a book of tables !
Thank you !

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