gives Owen's T function .


  • Mathematical function, suitable for both symbolic and numerical evaluation.
  • for real .
  • OwenT[x,a] is an entire function of x with no branch cut discontinuities.
  • OwenT[x,a] has a branch cut discontinuity in the complex a plane running from to and from to .
  • For certain special arguments, OwenT automatically evaluates to exact values.
  • OwenT can be evaluated to arbitrary numerical precision.
  • OwenT automatically threads over lists.


open allclose all

Basic Examples  (6)

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:

Series expansion at a singular point:

Scope  (30)

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)

Values of OwenT at fixed points:

OwenT for symbolic a:

Values at zero:

Find the first positive maximum of OwenT[x,1 ]:

Compute the associated OwenT[x,1] function:

Visualization  (3)

Plot the OwenT function for various parameters:

Plot the real part of :

Plot the imaginary part of :

Plot as real parts of two parameters vary:

Function Properties  (11)

OwenT is defined for all real values:

is defined for :

is even with respect to and odd with respect to :

OwenT may reduce to a simpler form:

OwenT is an analytic function:

OwenT is neither non-decreasing nor non-increasing:

is not injective for :

is not surjective for :

is non-negative for :

OwenT has no singularities or discontinuities:

OwenT is neither convex nor concave:

TraditionalForm formatting:

Differentiation  (2)

First derivative with respect to x:

First derivative with respect to a:

Higher derivatives with respect to x:

Plot the higher derivatives with respect to x when a=1.5:

Integration  (3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

Series Expansions  (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

Applications  (5)

Plot Owen's T-function in the complex a plane:

Compute the CDF of SkewNormalDistribution:

Compute the probability of an uncorrelated bivariate normal over a truncated wedge:

The probability for a standard binormal variate with correlation to lie inside a triangle can be expressed using OwenT:

Visualize the region:

Evaluate the probability for a particular value of the correlation coefficient:

Use NProbability to compute the probability directly:

Use OwenT to compute the standard BinormalDistribution probability of :

Evaluate numerically:

Compute directly:

Properties & Relations  (1)


Wolfram Research (2010), OwenT, Wolfram Language function, https://reference.wolfram.com/language/ref/OwenT.html.


Wolfram Research (2010), OwenT, Wolfram Language function, https://reference.wolfram.com/language/ref/OwenT.html.


Wolfram Language. 2010. "OwenT." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/OwenT.html.


Wolfram Language. (2010). OwenT. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/OwenT.html


@misc{reference.wolfram_2022_owent, author="Wolfram Research", title="{OwenT}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/OwenT.html}", note=[Accessed: 01-December-2022 ]}


@online{reference.wolfram_2022_owent, organization={Wolfram Research}, title={OwenT}, year={2010}, url={https://reference.wolfram.com/language/ref/OwenT.html}, note=[Accessed: 01-December-2022 ]}