In this post I will show you how to calculate and plot the Spehrical Bessel Functions () of the first kind using C and Gnuplot.

We will use the following information:

and the recurrence relation:

We will create a program that calculates the values of the Bessel Function at various z values and for different n and store these values in a txt file. Then just plot it using Gnuplot.

We will create two functions called ‘b0’ and ‘b1’, that contain the definition of respectively.

Then we will create a function ‘bn’ that will use the first two functions and recursion to find the value of Bessel function for different z,n.

NOTE: I am using a slightly modified form of the recurrence relation. To get the form I am using, just replace n by n-1.

### C PROGRAM:

/*********************************************** **********SPHERICAL BESSEL FUNCTIONS************ ***********************************************/ #include<stdio.h> #include<math.h> /*Define j0(z) */ double b0(double z){ return sin(z)/z; } /*Define j1(z) */ double b1(double z){ return sin(z)/(z*z)-cos(z)/z; } /*Define jn(z) */ double bn(double z,int n){ double out; if (n==0){ out = b0(z); } else if(n==1){ out = b1(z); } /*using recurrence relation */ else{ out = (2*n-1)*bn(z,n-1)/z-bn(z,n-2); } return out; } main(){ double z; int n; FILE *fp=NULL; fp=fopen("bessel.txt","w"); for(z=0.01;z<=20;z=z+0.01){ //fprintf(fp,"%lf\t%lf\n",z,bn(z,3)); fprintf(fp,"%lf\t%lf\t%lf\t%lf\t%lf\t%lf\t%lf\n",z,bn(z,0),bn(z,1),bn(z,2),bn(z,3),bn(z,4),bn(z,5)); } }

When you run the above C, it will generate a file called ‘bessel.txt’ which would contain 7 columns of data-points.

The first column contains the ‘z’ values and the rest of them are for

These can be easily plotted using Gnuplot by using the following commands:

### Gnuplot Command:

`->set xlabel "z"`

`->plot 'bessel.txt' u 1:2 w l t "j0(z)", '' u 1:3 w l t "j1(z)", '' u 1:4 w l t "j2(z)", '' u 1:5 w l t "j3(z)", '' u 1:6 w l t "j4(z)", '' u 1:7 w l t "j5(z)"`

### OUTPUT (Gnuplot):

### YouTube Tutorial:

Ph.D. researcher at Friedrich-Schiller University Jena, Germany. I’m a physicist specializing in computational material science. I write efficient codes for simulating light-matter interactions at atomic scales. I like to develop Physics, DFT, and Machine Learning related apps and software from time to time. Can code in most of the popular languages. I like to share my knowledge in Physics and applications using this Blog and a YouTube channel.