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OwenT

OwenT[x,a]

gives Owen's T function .

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 (38)

Numerical Evaluation (6)

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:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix OwenT function using MatrixFunction:

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 the real part of the function as 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 in both its arguments:

It is not an analytic function over the complexes:

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:

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 :

Ordinary differential equation with respect to satisfied by :

Partial differential equation satisfied by :

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. :

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