HP Prime, Casio Classpad: Bessell Functions of the 1st Kind

Bessel Function of the First Kind

The program BESS1 calculates the Bessel function of the First Kind. With the HP Prime and Casio Classpad, and any calculators that can process integrals (and the processor is fast enough), we can use the definition:

J_n(x) = 1/π * ∫ (cos(n*t – x*sin(t)) dt for t= 0 to t = π

This is the solution to the differential equation:

x^2*y’’ + x*y’ + (x^2 – n^2)*y = 0, where y is a function of x.

There are many other variations and approximations to calculate J_n(x) if the integral function is not available on your calculator. 

More information on the Bessel function is found here:

Wolfram

http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

Wikipedia

https://en.wikipedia.org/wiki/Bessel_function

Casio Classpad (fx-CP400):  The Function BESS1

As the screen above shows, I used a Define statement in the Main Mode:

Define bess1(n,x) = 1/π ∫ (cos(n*t – x*sin(t)) dt, from t = 0 to t = π)

Note:  Make sure that the calculator is in Radians mode before calculating.

HP Prime:  BESS1

EXPORT BESS1(n,t)

BEGIN

// Bessel 1st Kind

LOCAL b;

// Integrate

b:=(1/π)*CAS.int(COS(n*X-t*SIN(X)),X,0,π);

// Approximate

b:=approx(b);

RETURN b;

END;

Note: the integration and approximation are two separate steps.  They need to be in order for this program to work correctly.

Examples

bess1(1,2) ≈ 0.576724807756

bess1(0,6.3) ≈ 0.223812006132

bess1(2,4) ≈ 0.364128145852

This blog is property of Edward Shore, 2016.