HypergeometricPFQRegularized

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`HypergeometricPFQRegularized`

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HypergeometricPFQRegularized[{a₁,…,a_p},{b₁,…,b_q},z]

is the regularized generalized hypergeometric function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
HypergeometricPFQRegularized is finite for all finite values of its arguments so long as .
For certain special arguments, HypergeometricPFQRegularized automatically evaluates to exact values.
HypergeometricPFQRegularized can be evaluated to arbitrary numerical precision.
HypergeometricPFQRegularized can be used with Interval and CenteredInterval objects. »

Examples

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

Evaluate numerically:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-v0k0a

Out[1]=1

Evaluate symbolically:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-dzpoz

Out[1]=1

Plot over a subset of the reals:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-m51

Out[1]=1

Plot over a subset of the complexes:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-kiedlx

Out[1]=1

Series expansion at the origin:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-f65ufv

Out[1]=1

Series expansion at Infinity:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-fgrnr3

Out[1]=1

Scope (33)Survey of the scope of standard use cases

Numerical Evaluation (6)

Evaluate numerically:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-l274ju

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-cksbl4

Out[2]=2

Evaluate to high precision:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-b0wt9

Out[1]=1

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

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-y7k4a

Out[2]=2

Complex number inputs:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-hfml09

Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-di5gcr

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-bq2c6r

Out[2]=2

HypergeometricPFQRegularized can be used with Interval and CenteredInterval objects:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-cmdnbi

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-ddeque

Out[2]=2

Compute the elementwise values of an array:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-thgd2

Out[1]=1

Or compute the matrix HypergeometricPFQRegularized function using MatrixFunction:

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-o5jpo

Out[2]=2

Specific Values (4)

For simple parameters, HypergeometricPFQRegularized evaluates to simpler functions:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-o7spmt

Out[1]=1

Evaluate symbolically:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-jql9vr

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-h2sg3

Out[2]=2

Value at zero:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-jevg27

Out[1]=1

Find a value of for which HypergeometricPFQRegularized[{2,1},{2,3},x]1.5:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-f2hrld

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-e82ip3

Out[2]=2

Visualization (2)

Plot the HypergeometricPFQRegularized function for various parameters:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-c0x9p4

Out[1]=1

Plot the real part of :

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-kgd8nu

Out[1]=1

Plot the imaginary part of :

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-cvjvsr

Out[2]=2

Function Properties (10)

HypergeometricPFQRegularized is defined for all real and complex values:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-cl7ele

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-de3irc

Out[2]=2

HypergeometricPFQRegularized threads elementwise over lists in its third argument:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-gtl3ub

Out[1]=1

HypergeometricPFQRegularized is an analytic function of z for specific values:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-dpcmbc

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-1dshm

Out[2]=2

HypergeometricPFQRegularized is neither non-decreasing nor non-increasing for specific values:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-rmb7f

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-elzs35

Out[2]=2

HypergeometricPFQRegularized[{1,1,1},{3,3,3},z] is injective:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-g54sqh

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-zf7zy

Out[2]=2

HypergeometricPFQRegularized[{1,1,1},{3,3,3},z] is not surjective:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-klmhpu

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-b5ts4n

Out[2]=2

HypergeometricPFQRegularized is neither non-negative nor non-positive:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-bdbh0f

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-codo9f

Out[2]=2

HypergeometricPFQRegularized[{1,1,2},{3,3},z] has both singularity and discontinuity for z≥1 and at zero:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-hl8oqu

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-counuv

Out[2]=2

HypergeometricPFQRegularized is neither convex nor concave:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-ci2sbr

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-fybize

Out[2]=2

TraditionalForm formatting:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-cegk5j

Differentiation (3)

First derivative with respect to :

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-krpoah

Out[1]=1

Higher derivatives with respect to :

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-z33jv

Out[1]=1

Plot the higher derivatives with respect to when and :

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-csvztd

Out[2]=2

Formula for the derivative with respect to z when a₁=1,a₂=2 and b₁=b₂=b₃=3:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-cb5zgj

Out[1]=1

Integration (3)

Compute the indefinite integral using Integrate:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-bponid

Out[1]=1

Verify the antiderivative:

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-op9yly

Out[2]=2

Definite integral:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-bfdh5d

Out[1]=1

More integrals:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-4nbst

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-yncg8

Out[2]=2

Series Expansions (5)

Find the Taylor expansion using Series:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-ewr1h8

Out[1]=1

Plots of the first three approximations around :

In[3]:=3

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https://wolfram.com/xid/040178ialp5dpzgy-binhar

Out[4]=4

General term in the series expansion using SeriesCoefficient:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-dznx2j

Out[1]=1

Find the series expansion at Infinity:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-syq

Out[1]=1

Find the series expansion for an arbitrary symbolic direction :

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-t5t

Out[1]=1

Taylor expansion at a generic point:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-jwxla7

Out[1]=1

Applications (1)Sample problems that can be solved with this function

Find a fractional derivative of BesselJ:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-53iy04

Out[1]=1

Integral of order of BesselJ[0,z]:

In[2]:=2

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https://wolfram.com/xid/040178ialp5dpzgy-khd5qe

Out[2]=2

Properties & Relations (2)Properties of the function, and connections to other functions

Use FunctionExpand to express the input in terms of simpler functions:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-dzpt0

Out[1]=1

Integrate may return results involving HypergeometricPFQRegularized:

In[1]:=1

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https://wolfram.com/xid/040178ialp5dpzgy-deb13z

Out[1]=1

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