4.8 Advanced Topic: A Few Examples of Special Functions
Some special functions.
Gamma Function
For a positive integer 
, 
. 
can be viewed as a generalization of the factorial function, valid for complex arguments 
. The Euler gamma function Gamma[z] is defined by the integral 
.
Mathematica TE gives exact results for some values of special functions.
In[1]:= Gamma[15/2]
Out[1]= 
No exact result is known here.
In[2]:= Gamma[15/7]
Out[2]= 
A numerical result, to arbitrary precision, can nevertheless be found.
In[3]:= N[%, 40]
Out[3]= 
You can give complex arguments to special functions.
In[4]:= N[ Gamma[3 + 4I] ]
Out[4]= 
Special functions automatically get applied to each element in a list.
In[5]:= Gamma[{3/2, 5/2, 7/2}]
Out[5]= 
Here is a contour plot of the gamma function in the complex plane.
In[6]:= ContourPlot[ Abs[Gamma[x + I y]], {x, -3, 3},
{y, -2, 2}, PlotPoints -> 40 ]
Out[6]= 
Error Function
The error function Erf[z] is related to the integral of the Gaussian distribution, given by 
. A common use of the error function is to evaluate the integral 
. This is given by Erf[w, z].
Bessel Functions
The Bessel functions BesselJ[n, z] and BesselY[n, z] are linearly independent solutions to the differential equation 
.
Bessel functions arise in solving differential equations for systems with cylindrical symmetry.
is often called the Bessel function of the first kind, or simply the Bessel function. 
is referred to as the Bessel function of the second kind, the Weber function, or the Neumann function (denoted 
).
You can use FindRoot to find roots of special functions.
In[7]:= FindRoot[ BesselJ[0, x], {x, 1} ]
Out[7]= 
Zeta Function
The Riemann zeta function Zeta[s] is defined by the relation 
(for 
). Zeta functions with integer arguments arise in evaluating various sums and integrals. Mathematica TE gives exact results when possible for zeta functions with integer arguments.
It is possible to extend the definition of 
to arbitrary complex 
. The zeta function for complex arguments is central to number-theoretical studies of the distribution of primes. Of particular importance are the values about the critical line 
.
Mathematica TE gives exact results for 
.
In[8]:= Zeta[6]
Out[8]= 
Here is a three-dimensional picture of the Riemann zeta function in the complex plane.
In[9]:= Plot3D[ Abs[ Zeta[x + I y] ], {x, -3, 3},
{y, 2, 35}, PlotPoints -> 30 ]
Out[9]= 
Other Mathematica function related to the factorial function. (See Section 3.2.)
Other Mathematica special functions, related to the hypergeometric functions. (See Section 3.2.)
Other Mathematica special functions, related to elliptic functions. (See Section 3.2.)
