CarlsonRF

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`CarlsonRF`

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CarlsonRF[x,y,z]

gives the Carlson's elliptic integral .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
For , and , $TemplateBox[{x, y, z}, CarlsonRF]⩵1/2int_0^infty(t+x)^(-1/2)(t+y)^(-1/2)(t+z)^(-1/2)dt$ .
CarlsonRF[x,y,z] has a branch cut discontinuity for .
For certain arguments, CarlsonRF automatically evaluates to exact values.
CarlsonRF can be evaluated to arbitrary numerical precision.
CarlsonRF automatically threads over lists.
CarlsonRF can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples (3)Summary of the most common use cases

Evaluate numerically:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-dpixqm

Out[1]=1

Plot over a range of arguments:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-gi0fk3

Out[1]=1

CarlsonRF is related to the Legendre elliptic integral of the first kind for :

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-nrznwy

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bnhv6qov8gai-vp8qn

Out[2]=2

Scope (17)Survey of the scope of standard use cases

Numerical Evaluation (6)

Evaluate numerically:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-ca7jx4

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bnhv6qov8gai-msin0h

Out[2]=2

Evaluate at high precision:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-c9glx4

Out[1]=1

Precision of the output tracks the precision of the input:

In[2]:=2

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https://wolfram.com/xid/0bnhv6qov8gai-dryelq

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0bnhv6qov8gai-blyyy3

Out[3]=3

Evaluate for complex arguments:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-csaoq3

Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-bnknmr

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bnhv6qov8gai-73gnj

Out[2]=2

CarlsonRF threads elementwise over lists:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-cn7alx

Out[1]=1

CarlsonRF can be used with Interval and CenteredInterval objects:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-dwg06f

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bnhv6qov8gai-hlryvn

Out[2]=2

Specific Values (4)

Simple exact results are generated automatically:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-eftyik

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bnhv6qov8gai-gdebdv

Out[2]=2

When one argument of CarlsonRF is zero, CarlsonRF reduces to the complete elliptic integral CarlsonRK:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-ns44x5

Out[1]=1

When two of the arguments of CarlsonRF are identical and do not lie on the negative real axis, CarlsonRF reduces to CarlsonRC:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-ndydra

Out[1]=1

When all arguments of CarlsonRF are identical and do not lie on the negative real axis, CarlsonRF reduces to an elementary function:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-buy3cp

Out[1]=1

Differentiation and Integration (2)

Derivative of CarlsonRF is proportional to CarlsonRD:

In[2]:=2

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https://wolfram.com/xid/0bnhv6qov8gai-kgaxsf

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0bnhv6qov8gai-dhz88v

Out[3]=3

Indefinite integral of CarlsonRF:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-kqp6bj

Out[1]=1

Function Representations (1)

TraditionalForm formatting:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-bdxsaz

Function Identities and Simplifications (4)

An equation relating CarlsonRF, CarlsonRG and CarlsonRD:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-ejtp1i

Out[1]=1

CarlsonRF satisfies the Euler–Poisson partial differential equation:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-fpunj

Out[1]=1

CarlsonRF satisfies Euler's homogeneity relation:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-zxe9v

Out[1]=1

A partial differential equation satisfied by CarlsonRF:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-n1rgg3

Out[1]=1

Applications (3)Sample problems that can be solved with this function

Distance along a meridian of the Earth:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-hee91

Out[1]=1

Compare with the result of GeoDistance:

In[2]:=2

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https://wolfram.com/xid/0bnhv6qov8gai-di7s2e

Out[2]=2

Expectation value of the reciprocal square root of a quadratic form over a normal distribution:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-f0ryoc

In[2]:=2

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https://wolfram.com/xid/0bnhv6qov8gai-fkufen

Out[2]=2

Compare with the closed-form result in terms of CarlsonRF:

In[3]:=3

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https://wolfram.com/xid/0bnhv6qov8gai-co4nqb

Out[3]=3

Express EllipticLog in terms of CarlsonRF:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-er1fbx

Out[1]=1

Properties & Relations (1)Properties of the function, and connections to other functions

CarlsonRF is invariant under a permutation of its arguments:

In[1]:=1

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https://wolfram.com/xid/0bnhv6qov8gai-hqjxwc

Out[1]=1

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