# Special functions

Special functions

as they arise in Mathematica

### The error function

The error function

The normal distribution is a function of tremendous importance in probability and statistics. Geometrically, it represents the so-called bell-shaped curve.

In[]:=

normal[x_]=Exp[-/2]

2

x

2π

;bellCurve=Plot[normal[x],{x,-4,4},AspectRatio1/3,ImageSize600,PlotStyleBlack]Out[]=

An important property of this thing is that the total area under the curve is one. Of course, we can check this via integration:

In[]:=

Integrate[normal[x],{x,-Infinity,Infinity}]

Out[]=

1

Normal probabilities, say can be expressed as an area under the graph of the normal distribution and over an interval on the real axis. Thus, we need to be able to compute areas like so:

P(0<Z<1.5)

Out[]=

Again, we should be able to do this via integration but we get a weird response when we try Mathematica’s Integrate command:

In[]:=

Integrate[normal[x],x]

Out[]=

1

2

x

2

According to the Documentation, this Erf thing is exactly the value of the following integral:

In[]:=

2Integrate[Exp[-],{t,0,x}]

2

t

π

Out[]=

Erf[x]

In fact, that’s the whole point! We can always use numerical integration to get estimates to normal probabilities:

In[]:=

NIntegrate[normal[x],{x,0,3/2}]

Out[]=

0.433193

### Bessel functions

Bessel functions

Here’s an ODE that we’ll meet soon enough:

In[]:=

DSolve[u''[r]+u'[r]/r-λu[r],u[r],r]

Out[]=

u[r]BesselJ0,r+BesselY0,r

λ

1

λ

2

The solution is expressed in terms of special functions called Bessel functions.

In[]:=

DSolve[y''[x]+xy'[x]+(-)y[x]0,y[x],x]

2

x

2

x

2

a

Out[]=

{{y[x]BesselJ[a,x]+BesselY[a,x]}}

1

2

In[]:=

Manipulate[Plot[BesselJ[a,x],{x,-2,T},PlotRangeAll,ImageSize600],{{a,0},Range[0,3]},{{T,12},1,42}]

Out[]=