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LameC
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LameC

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LameC[ν,j,z,m]

gives the ^(th) Lamé function TemplateBox[{nu, j, z, m}, LameC] of order with elliptic parameter .

Details

  • LameC belongs to the Lamé class of functions and solves boundary-value problems for Laplace's equation in ellipsoidal and spheroconal coordinates, as well as occurring in other problems of mathematical physics and quantum mechanics.
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • LameC[ν,j,z,m] satisfies the Lamé differential equation , with the Lamé eigenvalue given by LameEigenvalueA[ν,j,m], and where TemplateBox[{z, m}, JacobiSN] is the Jacobi elliptic function JacobiSN[z,m].
  • For certain special arguments, LameC automatically evaluates to exact values.
  • LameC can be evaluated to arbitrary numerical precision for an arbitrary complex argument.
  • LameC automatically threads over lists.
  • LameC[ν,0,z,0]= and LameC[ν,j,z,0]=Cos[j(-z)].
  • LameC[ν,j,z,m] is proportional to HeunG[a,q,α,β,γ,δ,ξ], where , if the parameters of HeunG are specialized as follows: .

Examples

open allclose all

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

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-3h5x4e
Out[1]=1

Plot the LameC function for and :

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-7ugmrr
Out[1]=1

Series expansion of LameC at the origin:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-z8evs7
Out[1]=1

Scope  (27)Survey of the scope of standard use cases

Numerical Evaluation  (5)

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-okwtpc
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-kkvikx
Out[1]=1

LameC can take complex number parameters and argument:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-hqtw40
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0wgtmv5-zgyi26
Out[2]=2

Evaluate LameC efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-qn20jy
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgtmv5-jlo2be
Out[2]=2

Lists and matrices:

In[1]:=1
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https://wolfram.com/xid/0wgtmv5-clqbry
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgtmv5-dhyb88
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0wgtmv5-h6iizh
Out[3]=3

Specific Values  (3)

Value of LameC when and :

In[1]:=1
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https://wolfram.com/xid/0wgtmv5-vlsc9t
Out[1]=1

Value of LameC when and :

In[2]:=2
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https://wolfram.com/xid/0wgtmv5-i8zbem
Out[2]=2

Some poles of LameC:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-6x592
Out[1]=1

For integer values of and , LameC can be expressed entirely in terms of Jacobi elliptic functions:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-by033o
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgtmv5-5h6j9
Out[2]=2

Visualization  (6)

Plot the first three even LameC functions:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-2nkt2z
Out[1]=1

Plot the first three odd LameC functions:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-eezd2
Out[1]=1

Plot the absolute value of the LameC function for complex parameters:

In[1]:=1
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https://wolfram.com/xid/0wgtmv5-cv7enz
Out[1]=1

Plot LameC as a function of its first parameter :

In[1]:=1
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https://wolfram.com/xid/0wgtmv5-0oazxn
Out[1]=1

Plot LameC as a function of and elliptic parameter :

In[1]:=1
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https://wolfram.com/xid/0wgtmv5-kavekp
Out[1]=1

Plot the family of LameC functions for different values of the elliptic parameter :

In[1]:=1
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https://wolfram.com/xid/0wgtmv5-m5b5wf
Out[1]=1

Function Properties  (2)

When is even, LameC is a periodic function of real argument with a period 2EllipticK[m]:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-20emv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgtmv5-pbglku
Out[2]=2

When is odd, LameC is a periodic function of real argument with a period 4EllipticK[m] and has an initial value LameC[ν,j,0,m]=0:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-m4hob1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgtmv5-hvapl9
Out[2]=2

Differentiation  (3)

The -derivative of LameC is LameCPrime:

In[1]:=1
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https://wolfram.com/xid/0wgtmv5-6eb2k6
Out[1]=1

Higher derivatives of LameC are calculated using LameCPrime:

In[1]:=1
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https://wolfram.com/xid/0wgtmv5-baz2gr
Out[1]=1

Derivatives of LameC for specific cases of parameters:

In[1]:=1
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https://wolfram.com/xid/0wgtmv5-6je6g8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgtmv5-yw3970
Out[2]=2

Integration  (3)

Indefinite integrals of LameC cannot be expressed in elementary or other special functions:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-ecaem6
Out[1]=1

Definite numerical integrals of LameC:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-yy7qm6
Out[1]=1

More integrals with LameC:

In[1]:=1
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https://wolfram.com/xid/0wgtmv5-77lshe
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0wgtmv5-d5fnzu
Out[2]=2

Series Expansions  (3)

Series expansion of LameC at the origin:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-0rpbb6
Out[1]=1

Coefficient of the second-order term of this expansion:

In[2]:=2
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https://wolfram.com/xid/0wgtmv5-9rxgh1
Out[2]=2

Plot the first- and third-order approximations for LameC around :

In[1]:=1
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https://wolfram.com/xid/0wgtmv5-zgnxgy
In[2]:=2
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https://wolfram.com/xid/0wgtmv5-jch4ka
In[3]:=3
✖
https://wolfram.com/xid/0wgtmv5-hrtnwe
Out[3]=3

Series expansion for LameC at any ordinary complex point:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-ukhgue
Out[1]=1

Function Representations  (2)

LameC cannot be represented in terms of MeijerG:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-bfvi7o
Out[1]=1

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-ha9m21

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

LameC solves the Lamé differential equation when h=LameEigenvalueA[ν,j,m]:

In[11]:=11
✖
https://wolfram.com/xid/0wgtmv5-nkp6ld
In[14]:=14
✖
https://wolfram.com/xid/0wgtmv5-8yj5vx
Out[14]=14

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

LameC is an even function when is a non-negative even integer:

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-2dos
Out[1]=1

LameC is an odd function when is a positive odd integer:

In[2]:=2
✖
https://wolfram.com/xid/0wgtmv5-1317ex
Out[2]=2

Use FunctionExpand to expand LameC for integer values of and :

In[1]:=1
✖
https://wolfram.com/xid/0wgtmv5-bxtg9f
Out[1]=1

Possible Issues  (1)Common pitfalls and unexpected behavior

LameC is not defined if is a negative integer:

In[2]:=2
✖
https://wolfram.com/xid/0wgtmv5-xzpq1j
Out[2]=2

LameC is not defined if is not an integer:

In[3]:=3
✖
https://wolfram.com/xid/0wgtmv5-n1xueg
Out[3]=3
Wolfram Research (2020), LameC, Wolfram Language function, https://reference.wolfram.com/language/ref/LameC.html.
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Wolfram Research (2020), LameC, Wolfram Language function, https://reference.wolfram.com/language/ref/LameC.html.

Text

Wolfram Research (2020), LameC, Wolfram Language function, https://reference.wolfram.com/language/ref/LameC.html.

✖
Wolfram Research (2020), LameC, Wolfram Language function, https://reference.wolfram.com/language/ref/LameC.html.

CMS

Wolfram Language. 2020. "LameC." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LameC.html.

✖
Wolfram Language. 2020. "LameC." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LameC.html.

APA

Wolfram Language. (2020). LameC. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LameC.html

✖
Wolfram Language. (2020). LameC. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LameC.html

BibTeX

@misc{reference.wolfram_2025_lamec, author="Wolfram Research", title="{LameC}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/LameC.html}", note=[Accessed: 11-July-2025 ]}

✖
@misc{reference.wolfram_2025_lamec, author="Wolfram Research", title="{LameC}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/LameC.html}", note=[Accessed: 11-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_lamec, organization={Wolfram Research}, title={LameC}, year={2020}, url={https://reference.wolfram.com/language/ref/LameC.html}, note=[Accessed: 11-July-2025 ]}

✖
@online{reference.wolfram_2025_lamec, organization={Wolfram Research}, title={LameC}, year={2020}, url={https://reference.wolfram.com/language/ref/LameC.html}, note=[Accessed: 11-July-2025 ]}