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OwenT
✖
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

open allclose all

Basic Examples  (6)Summary of the most common use cases

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-jqirte
Out[1]=1

Plot over a subset of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-bi2uyi
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-eubfov
Out[2]=2

Plot over a subset of the complexes:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-kiedlx
Out[1]=1

Series expansion at the origin:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-t1nfp
Out[1]=1

Series expansion at Infinity:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-cugjvu
Out[1]=1

Series expansion at a singular point:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-iffsoc
Out[1]=1

Scope  (38)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-cksbl4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-wlv0g
Out[2]=2

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-b0wt9
Out[1]=1

The precision of the output tracks the precision of the input:

In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-y7k4a
Out[2]=2

Complex number input:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-hfml09
Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-di5gcr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-bq2c6r
Out[2]=2

Compute average-case statistical intervals using Around:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-cw18bq
Out[1]=1

Compute the elementwise values of an array:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-thgd2
Out[1]=1

Or compute the matrix OwenT function using MatrixFunction:

In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-o5jpo
Out[2]=2

Specific Values  (5)

Values of OwenT at fixed points:

In[1]:=1
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https://wolfram.com/xid/0cg5yhtbv-nww7l
Out[1]=1

OwenT for symbolic a:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-fc9m8o
Out[1]=1

Values at zero:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-bmqd0y
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-f2hrld
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-cc4pqr
Out[2]=2

Compute the associated OwenT[x,1] function:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-klij8s
Out[1]=1

Visualization  (3)

Plot the OwenT function for various parameters:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-ecj8m7
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-beyki2
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-zpr05
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-elqrq8
Out[1]=1

Function Properties  (11)

OwenT is defined for all real values:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-cl7ele
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-de3irc
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-dnla5q
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-bign6j
Out[2]=2

OwenT may reduce to a simpler form:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-c1zts4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-lk64xf
Out[2]=2

OwenT is an analytic function in both its arguments:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-h5x4l2
Out[1]=1

It is not an analytic function over the complexes:

In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-vuist
Out[2]=2

OwenT is neither non-decreasing nor non-increasing:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-rmb7f
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-8nsw5z
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-zf7zy
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-wybzkl
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-c5tkuy
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-oofpk0
Out[1]=1

OwenT has no singularities or discontinuities:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-ho029y
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-c20lli
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cg5yhtbv-2gvuj
Out[3]=3

OwenT is neither convex nor concave:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-duxck
Out[1]=1

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-biam2o

Differentiation  (4)

First derivative with respect to x:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-krpoah
Out[1]=1

Higher derivatives with respect to x:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-z33jv
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-fxwmfc
Out[2]=2

First derivative with respect to a:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-dhgaw2
Out[1]=1

Higher derivatives with respect to a:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-nx5bsu
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-c1xxhb
Out[2]=2

Integration  (4)

Compute the indefinite integral with respect to using Integrate:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-bponid
Out[1]=1

Verify the antiderivative:

In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-op9yly
Out[2]=2

Compute the indefinite integral with respect to :

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-c4ni28
Out[1]=1

Verify the antiderivative:

In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-23a5n
Out[2]=2

Definite integral:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-bfdh5d
Out[1]=1

More integrals:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-4nbst
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-yncg8
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cg5yhtbv-bf5nsh
Out[3]=3

Series Expansions  (2)

Find the Taylor expansion using Series:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-ewr1h8
Out[1]=1

Plots of the first three approximations around :

In[3]:=3
✖
https://wolfram.com/xid/0cg5yhtbv-binhar
Out[4]=4

Taylor expansion at a generic point:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-jwxla7
Out[1]=1

Function Identities  (3)

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-c9s5wf
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-h6kqn1
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-rb4u3
Out[1]=1

Applications  (6)Sample problems that can be solved with this function

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-bzo6k
Out[1]=1

Compute the CDF of SkewNormalDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-ek7p0p
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-cwn3p3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-e8iup0
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-cs7c0w

Generate and visualize the region:

In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-e5joza
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cg5yhtbv-comurd
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0cg5yhtbv-gd2npf
Out[4]=4

Use NProbability to compute the probability directly:

In[5]:=5
✖
https://wolfram.com/xid/0cg5yhtbv-bqbrn6
Out[5]=5

Use OwenT to compute the standard BinormalDistribution probability of :

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-r08oo

Evaluate numerically:

In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-y1741
Out[2]=2

Compute directly:

In[3]:=3
✖
https://wolfram.com/xid/0cg5yhtbv-hmisy8
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5yhtbv-fuyu0j
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0cg5yhtbv-3pn8al
Out[2]=2
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.

Text

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.

CMS

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. 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

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

BibTeX

@misc{reference.wolfram_2025_owent, author="Wolfram Research", title="{OwenT}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/OwenT.html}", note=[Accessed: 29-March-2025 ]}

✖
@misc{reference.wolfram_2025_owent, author="Wolfram Research", title="{OwenT}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/OwenT.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_owent, organization={Wolfram Research}, title={OwenT}, year={2010}, url={https://reference.wolfram.com/language/ref/OwenT.html}, note=[Accessed: 29-March-2025 ]}

✖
@online{reference.wolfram_2025_owent, organization={Wolfram Research}, title={OwenT}, year={2010}, url={https://reference.wolfram.com/language/ref/OwenT.html}, note=[Accessed: 29-March-2025 ]}