Function Repository Resource:

# ParabolicCylinderU

Evaluate the Weber parabolic cylinder function U

Contributed by: Jan Mangaldan
 ResourceFunction["ParabolicCylinderU"][a,z] gives the Weber parabolic cylinder function U(a,z).

## Details

Mathematical function, suitable for both symbolic and numerical manipulation.
U(a,z) satisfies the differential equation .
ResourceFunction["ParabolicCylinderU"][a,z] is an entire function of both a and z with no branch cut discontinuities.
For certain special arguments, ResourceFunction["ParabolicCylinderU"] automatically evaluates to exact values.
ResourceFunction["ParabolicCylinderU"] can be evaluated to arbitrary numerical precision.

## Examples

### Basic Examples (3)

Evaluate numerically:

 In[1]:=
 Out[1]=

Plot :

 In[2]:=
 Out[2]=

Series expansion at the origin:

 In[3]:=
 Out[3]=

### Scope (4)

Evaluate for complex arguments and parameters:

 In[4]:=
 Out[4]=

Evaluate to high precision:

 In[5]:=
 Out[5]=

The precision of the output tracks the precision of the input:

 In[6]:=
 Out[6]=

Simple exact input gives exact results:

 In[7]:=
 Out[7]=

 In[8]:=
 Out[8]=

### Properties and Relations (4)

ParabolicCylinderU satisfies the Weber differential equation:

 In[9]:=
 Out[9]=

A recurrence relation satisfied by ParabolicCylinderU:

 In[10]:=
 Out[10]=

Verify an expression for the derivative:

 In[11]:=
 Out[11]=

Express ParabolicCylinderU in terms of ParabolicCylinderV:

 In[12]:=
 Out[12]=

## Version History

• 1.0.0 – 17 May 2021