# OwenT

OwenT[x,a]

gives Owen's T function .

# Details • 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 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 :

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:

### 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. :