# ParabolicCylinderD

ParabolicCylinderD[ν,z]

gives the parabolic cylinder function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• satisfies the Weber differential equation .
• For certain special arguments, ParabolicCylinderD automatically evaluates to exact values.
• ParabolicCylinderD can be evaluated to arbitrary numerical precision.
• ParabolicCylinderD automatically threads over lists.
• ParabolicCylinderD[ν,z] is an entire function of z with no branch cut discontinuities.
• ParabolicCylinderD can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

## Scope(34)

### Numerical Evaluation(5)

Evaluate numerically:

Evaluate to high precision:

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

Complex number input:

Evaluate efficiently at high precision:

ParabolicCylinderD can be used with Interval and CenteredInterval objects:

### Specific Values(5)

ParabolicCylinderD for symbolic parameters:

Value at zero:

Limiting value at infinity:

Find the first positive maximum of ParabolicCylinderD:

Evaluate for half-integer parameters:

### Visualization(4)

Plot the ParabolicCylinderD function for integer ( ) and half-integer ( ) orders:

Plot the real part of :

Plot the imaginary part of :

Plot as real parts of two parameters vary:

Types 2 and 3 of ParabolicCylinderD function have different branch cut structures:

### Function Properties(10)

ParabolicCylinderD is defined for all real and complex values: is an analytic function of : is neither non-decreasing nor non-increasing for : is not injective for :

ParabolicCylinderD is not surjective: is neither non-negative nor non-positive for :

ParabolicCylinderD has no singularities or discontinuities: is neither convex nor concave for :

### Differentiation(3)

First derivative with respect to z:

Higher derivatives with respect to z

Plot the higher derivatives with respect to z when ν=2:

Formula for the  derivative with respect to z:

### Series Expansions(5)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

### Function Identities and Simplifications(2)

Function identity:

Recurrence identities:

## Generalizations & Extensions(2)

Series expansion for symbolic first argument:

Series expansion at infinity:

## Applications(2)

Find the solution of the Schrödinger equation for a quadratic oscillator for arbitrary energies:

ParabolicCylinderD solves the Weber equation:

## Properties & Relations(5)

Use FunctionExpand to expand ParabolicCylinderD into other functions:

Integrate expressions involving ParabolicCylinderD:

ParabolicCylinderD can be represented as a DifferentialRoot:

ParabolicCylinderD can be represented as a DifferenceRoot:

The exponential generating function for ParabolicCylinderD: