gives the parabolic cylinder function .


  • 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.


open allclose all

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  (26)

Numerical Evaluation  (4)

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:

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  (3)

ParabolicCylinderD is defined for all real and complex values::

ParabolicCylinderD threads elementwise over lists:

TraditionalForm formatting:

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 ^(th) 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  (1)

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

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:

Wolfram Research (2007), ParabolicCylinderD, Wolfram Language function,


Wolfram Research (2007), ParabolicCylinderD, Wolfram Language function,


@misc{reference.wolfram_2020_paraboliccylinderd, author="Wolfram Research", title="{ParabolicCylinderD}", year="2007", howpublished="\url{}", note=[Accessed: 11-May-2021 ]}


@online{reference.wolfram_2020_paraboliccylinderd, organization={Wolfram Research}, title={ParabolicCylinderD}, year={2007}, url={}, note=[Accessed: 11-May-2021 ]}


Wolfram Language. 2007. "ParabolicCylinderD." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2007). ParabolicCylinderD. Wolfram Language & System Documentation Center. Retrieved from