Exercise "plots"

• Scientific plots with gnuplot.
• Prerequisites: install gnuplot at your box,

  sudo apt install gnuplot
  

  For MacOS with homebrew the command is

  brew install gnuplot [--with-qt|--with-x11] [--with-cairo]
  

• Alternatively, you can install "plotutils" which include the (much simpler) "graph" utility,

  sudo apt install plotutils
  

Tasks

1. Plot the error-function together with several of its tabulated values (as a test: the curve should pass through the tabulated points).

   You can use the following approximation,

   static double erf(double x){
   /// single precision error function (Abramowitz and Stegun, from Wikipedia)
   if(x<0) return -erf(-x);
   double[] a={0.254829592,-0.284496736,1.421413741,-1.453152027,1.061405429};
   double t=1/(1+0.3275911*x);
   double sum=t*(a[0]+t*(a[1]+t*(a[2]+t*(a[3]+t*a[4]))));/* the right thing */
   return 1-sum*Exp(-x*x);
   } 

2. Plot the gamma-function together with several of its tabulated values (factorials) as a test. You can use the following [Stirling approximation],

   static double gamma(double x){
   ///single precision gamma function (Gergo Nemes, from Wikipedia)
   if(x<0)return PI/sin(PI*x)/gamma(1-x);
   if(x<9)return gamma(x+1)/x;
   double lngamma=x*Log(x+1/(12*x-1/x/10))-x+Log(2*PI/x)/2;
   return Exp(lngamma);
   }
   	

   Try to reproduce the plot from the Wikipedia article.

3. The gamma-function overflows easily, so the logarithm of the gamma function is often a more useful function,

   static double lngamma(double x){
   if(x<=0) throw new ArgumentException("lngamma: x<=0");
   if(x<9) return lngamma(x+1)-Log(x);
   return x*Log(x+1/(12*x-1/x/10))-x+Log(2*PI/x)/2;
   }
   	

   Using lngamma make a similar plot as above for the logarithm of the gamma-function.

4. (Extra) Implement the complex function complex G(complex z) that calculates Γ(z) for complex arguments: use the (suitable modified) formula from wikipedia above, it works for complex arguments as well, I believe. Try to reproduce the 3d plot of the absolute value of the Γ-function in the complex plane from Wikipedia article.