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Function Repository Resource:

 ParabolicCylinderU

Source Notebook

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.
ResourceFunction["ParabolicCylinderU"] automatically threads over lists.

Examples

Basic Examples (3) 

Evaluate numerically:

In[1]:=
ResourceFunction["ParabolicCylinderU"][1, 1.5]
Out[1]=

Plot :

In[2]:=
Plot[ResourceFunction["ParabolicCylinderU"][-7/2, x], {x, -6, 6}]
Out[2]=

Series expansion at the origin:

In[3]:=
Series[ResourceFunction["ParabolicCylinderU"][-7/2, x], {x, 0, 10}]
Out[3]=

Scope (4) 

Evaluate for complex arguments and parameters:

In[4]:=
ResourceFunction["ParabolicCylinderU"][2 + I, 1.5 - I]
Out[4]=

Evaluate to high precision:

In[5]:=
N[ResourceFunction["ParabolicCylinderU"][1, 7/2], 50]
Out[5]=

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

In[6]:=
ResourceFunction[ "ParabolicCylinderU"][1, 3.5000000000000000000000000000000]
Out[6]=

Simple exact input gives exact results:

In[7]:=
ResourceFunction["ParabolicCylinderU"][a, 0]
Out[7]=

ParabolicCylinderU threads elementwise over lists:

In[8]:=
ResourceFunction["ParabolicCylinderU"][{1, 2, 3}, 2.5]
Out[8]=

Properties and Relations (4) 

ParabolicCylinderU satisfies the Weber differential equation:

In[9]:=
y''[x] - (a + x^2/4) y[x] /. y -> Function[x, ResourceFunction["ParabolicCylinderU"][a, x]] // FullSimplify
Out[9]=

A recurrence relation satisfied by ParabolicCylinderU:

In[10]:=
x ResourceFunction["ParabolicCylinderU"][a, x] - ResourceFunction["ParabolicCylinderU"][a - 1, x] + (a + 1/2) ResourceFunction["ParabolicCylinderU"][a + 1, x] // FullSimplify
Out[10]=

Verify an expression for the derivative:

In[11]:=
D[ResourceFunction["ParabolicCylinderU"][a, x], x] == x/2 ResourceFunction["ParabolicCylinderU"][a, x] - ResourceFunction["ParabolicCylinderU"][a - 1, x] // FullSimplify
Out[11]=

Express ParabolicCylinderU in terms of ParabolicCylinderV:

In[12]:=
ResourceFunction["ParabolicCylinderU"][a, x] == \[Pi]/( Cos[a \[Pi]]^2 Gamma[ a + 1/2]) (ResourceFunction["ParabolicCylinderV"][a, -x] - Sin[a \[Pi]] ResourceFunction["ParabolicCylinderV"][a, x]) // FullSimplify
Out[12]=

Version History

  • 1.0.0 – 17 May 2021

Source Metadata

  • Citation:
    • "Parabolic Cylinder Functions." Chapter 12 in F. W. J. Olver et al. (eds.), NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/12. Release 1.1.4 of 2021-01-15.

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