OwenT

OwenT[x,a]

gives Owen's T function TemplateBox[{x, a}, OwenT].

Details

  • Mathematical function, suitable for both symbolic and numerical evaluation.
  • TemplateBox[{x, a}, OwenT]=1/(2pi)int_0^aexp(-x^2(1+t^2)/2)/(1+t^2)dt for real .
  • OwenT[x,a] is an entire function of x with no branch cut discontinuities.
  • OwenT[x,a] has branch cut discontinuities in the complex a plane running 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.

Examples

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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  (36)

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 TemplateBox[{z, 3}, OwenT]:

Plot the imaginary part of TemplateBox[{z, 3}, OwenT]:

Plot the real part of the function as two parameters vary:

Function Properties  (11)

OwenT is defined for all real values:

TemplateBox[{z, a}, OwenT] is defined for :

TemplateBox[{x, a}, OwenT] is even with respect to and odd with respect to :

OwenT may reduce to a simpler form:

OwenT is an analytic function in both its arguments:

It is not an analytic function over the complexes:

OwenT is neither non-decreasing nor non-increasing:

TemplateBox[{x, a}, OwenT] is not injective for :

TemplateBox[{x, a}, OwenT] is not surjective for :

TemplateBox[{x, a}, OwenT] is non-negative for :

OwenT has no singularities or discontinuities:

TemplateBox[{x, a}, OwenT] has branch cut discontinuities with respect to over the complexes:

OwenT is neither convex nor concave:

TraditionalForm formatting:

Differentiation  (4)

First derivative with respect to x:

Higher derivatives with respect to x:

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

First derivative with respect to a:

Higher derivatives with respect to a:

Plot the higher derivatives with respect to a when x=0.5:

Integration  (4)

Compute the indefinite integral with respect to using Integrate:

Verify the antiderivative:

Compute the indefinite integral with respect to :

Verify the antiderivative:

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:

Function Identities  (3)

Ordinary differential equation with respect to satisfied by TemplateBox[{x, a}, OwenT]:

Ordinary differential equation with respect to satisfied by TemplateBox[{x, a}, OwenT]:

Partial differential equation satisfied by TemplateBox[{x, a}, OwenT]:

Applications  (6)

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 that a standard binormal variate with correlation lies within an equilateral triangle can be expressed using OwenT:

Generate and 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:

Compute the mean residual life function of a skew-normal random variate:

Plot the mean residual life function for several values of parameter , including the limiting case of a normal variate, i.e. :

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_owent, author="Wolfram Research", title="{OwenT}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/OwenT.html}", note=[Accessed: 13-June-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_owent, organization={Wolfram Research}, title={OwenT}, year={2010}, url={https://reference.wolfram.com/language/ref/OwenT.html}, note=[Accessed: 13-June-2024 ]}