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LameCPrime

LameCPrime[ν,j,z,m]

gives the -derivative of the Lamé function of order with elliptic parameter .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
LameCPrime belongs to the Lamé class of functions.
For certain special arguments, LameCPrime automatically evaluates to exact values.
LameCPrime can be evaluated to arbitrary numerical precision for an arbitrary complex argument.
LameCPrime automatically threads over lists.
LameCPrime[ν,0,z,0]=0 and LameCPrime[ν,j,z,0]=j Sin[j(-z)].

Examples

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Basic Examples (3)

Evaluate numerically:

Wolfram Language code: LameCPrime[0.9, 2, 0.7, 0.3]

Plot the LameCPrime function for and :

Wolfram Language code:

With[{ν = 1.5, m = 0.5}, Plot[{LameCPrime[ν, 1, z, m], LameCPrime[ν, 2, z, m]}, {z, -2EllipticK[m], 2EllipticK[m]}, PlotLabels -> {"j = 1", "j = 2"}]]

Series expansion of LameCPrime at the origin:

Wolfram Language code: Series[LameCPrime[ν, j, z, m], {z, 0, 1}]

Scope (26)

Numerical Evaluation (5)

Evaluate to high precision:

Wolfram Language code: N[LameCPrime[9 / 10, 2, 7 / 10, 3 / 10], 50]

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

Wolfram Language code: LameCPrime[9 / 10, 2, 7 / 10, 0.33333333333333333333331]

LameCPrime can take complex number parameters and argument:

Wolfram Language code: LameCPrime[1.2 + I, 2, 0.7, 0.3]

Wolfram Language code: LameCPrime[1.2 + I, 2, 0.7 - 0.1I, 0.3 + I]

Evaluate LameCPrime efficiently at high precision:

Wolfram Language code: LameCPrime[0.9, 2, 0.7, 3 / 10`500]//Timing

Wolfram Language code: LameCPrime[0.9, 2, 0.7, 3 / 10`500 + I]//Timing

Lists and matrices:

Wolfram Language code: LameCPrime[0.9, 2, 0.7, {0.1, 0.1 + I, I, 4}]

Wolfram Language code: LameCPrime[0.9, 2, {0.1, 0.7, 0.1I, 4 - 1 / 2I}, 0.1]

Wolfram Language code:

LameCPrime[0.9, 2, 0.7, (|   |       |
| :- | :---- |
| π | u     |
| v | (π/2) |)]

Specific Values (3)

Value of LameCPrime when and :

Wolfram Language code: LameCPrime[ν, 0, z, 0]

Value of LameCPrime when and :

Wolfram Language code: LameCPrime[ν, 3, z, 0]

Some poles of LameCPrime:

Wolfram Language code: {LameCPrime[ν, 1, 3I EllipticK[1 - m], m], LameCPrime[ν, 1, 2EllipticK[m] + 3I EllipticK[1 - m], m]}

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

Wolfram Language code: LameCPrime[3, 0, z, (1/2)]//FunctionExpand

Wolfram Language code: LameCPrime[4, 2, z, (1/2)]//FunctionExpand

Visualization (6)

Plot the first three even LameCPrime functions:

Wolfram Language code:

With[{ν = 3 / 2, m = 1 / 2}, 
	Plot[{LameCPrime[ν, 2, z, m], LameCPrime[ν, 4, z, m], LameCPrime[ν, 6, z, m]}, {z, -2EllipticK[m], 2EllipticK[m]}]]

Plot the first three odd LameCPrime functions:

Wolfram Language code:

With[{ν = 3 / 2, m = 1 / 2}, 
	Plot[{LameCPrime[ν, 1, z, m], LameCPrime[ν, 3, z, m], LameCPrime[ν, 5, z, m]}, {z, -2EllipticK[m], 2EllipticK[m]}]]

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

Wolfram Language code: Plot[Abs[LameCPrime[3 / 2 + I, 3, z, 0.1 + 0.1I]], {z, -8EllipticK[1 / 3], 8EllipticK[1 / 3]}]

Plot LameCPrime as a function of its first parameter :

Wolfram Language code:

Plot[{LameCPrime[-1 / 2, 1, z, 9 / 10], LameCPrime[1, 1, z, 9 / 10], LameCPrime[3, 1, z, 9 / 10]}, {z, -4EllipticK[1 / 3], 4EllipticK[1 / 3]}]

Plot LameCPrime as a function of and elliptic parameter :

Wolfram Language code: Plot3D[LameCPrime[3 / 2, 1, z, m], {z, -5, 5}, {m, 0, 1}, ViewPoint -> {-1.3, 2.4, 2.}]

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

Wolfram Language code: Plot[Evaluate[Table[LameCPrime[3, 1, z, m], {m, 0, 9 / 10, 1 / 15}]], {z, -4EllipticK[1 / 3], 4EllipticK[1 / 3]}, ...]

Function Properties (2)

When is even, LameCPrime is a periodic function of real argument with a period 2EllipticK[m] and has an initial value LameCPrime[ν,j,0,m]=0:

Wolfram Language code:

With[{ν = 7 / 2, m = 9 / 10, j = 2}, 
	LameCPrime[ν, j, z + 2EllipticK[m], m]]

Wolfram Language code:

With[{ν = 7 / 2, m = 9 / 10, j = 2}, 
	Plot[LameCPrime[ν, j, z, m], {z, -2EllipticK[m], 2EllipticK[m]}, Epilog -> Point[{0, 0}]]]

When is odd, LameCPrime is a periodic function of real argument with a period 4EllipticK[m]:

Wolfram Language code:

With[{ν = 7 / 2, m = 9 / 10, j = 3}, 
	LameCPrime[ν, j, z + 4EllipticK[m], m]]

Wolfram Language code:

With[{ν = 7 / 2, m = 9 / 10, j = 3}, 
	Plot[LameCPrime[ν, j, z, m], {z, -4EllipticK[m], 4EllipticK[m]}]]

Differentiation (2)

The -derivative of LameCPrime involves LameC function:

Wolfram Language code: D[LameCPrime[ν, j, z, m], z]

Derivatives of LameCPrime for specific cases of parameters:

Wolfram Language code: D[LameCPrime[ν, 2, z, 0], z]

Wolfram Language code: D[LameCPrime[ν, 2, z, 0], {z, n}]

Integration (3)

Indefinite integral of LameCPrime is LameC:

Wolfram Language code: Integrate[LameCPrime[3 / 2, 2, z, 1 / 3], z]

Definite numerical integrals of LameCPrime:

Wolfram Language code: NIntegrate[LameCPrime[12 / 10, 2, z, 9 / 10], {z, 0, 1 / 3}]

More integrals with LameCPrime:

Wolfram Language code: NIntegrate[z^2LameCPrime[12 / 10, 2, z, 9 / 10], {z, -1, 1 / 3}]

Wolfram Language code: NIntegrate[Sin[z / 2]LameCPrime[12 / 10, 2, z, 9 / 10], {z, -1, 1 / 2}]

Series Expansions (3)

Series expansion of LameCPrime at the origin:

Wolfram Language code: Series[LameCPrime[ν, j, z, m], {z, 0, 1}]

Coefficient of the second term of this expansion:

Wolfram Language code: SeriesCoefficient[LameCPrime[ν, j, z, m], {z, 0, 2}]

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

Wolfram Language code: {ν, j, m} = {1 / 9, 1, 9 / 10};

Wolfram Language code: terms = Normal@Table[Series[LameCPrime[ν, j, z, m], {z, 0, l}], {l, 1, 3, 2}];

Wolfram Language code:

Plot[{LameCPrime[ν, j, z, m], terms}//Flatten//Evaluate, {z, -2EllipticK[m], 2EllipticK[m]}, PlotLegends -> {LameCPrime[ν, j, z, m], "1st approximation", "3rd approximation"}]

Series expansion for LameCPrime at any ordinary complex point:

Wolfram Language code: Series[LameCPrime[ν, j, z, m], {z, 1 / 2, 1}]

Function Representations (2)

LameCPrime cannot be represented in terms of MeijerG:

Wolfram Language code: MeijerGReduce[LameCPrime[ν, j, z, m], z]

TraditionalForm formatting:

Wolfram Language code: LameCPrime[ν, j, z, m]//TraditionalForm

Applications (1)

Use the LameCPrime function to calculate the derivatives of LameC:

Wolfram Language code: D[LameC[ν, j, z, m], {z, 3}]

Properties & Relations (2)

LameCPrime is an even function when is a positive odd integer:

Wolfram Language code: LameCPrime[ν, 1, -z, m]

LameCPrime is an odd function when is a non-negative even integer:

Wolfram Language code: LameCPrime[ν, 2, -z, m]

Use FunctionExpand to expand LameCPrime for integer values of and :

Wolfram Language code: LameCPrime[7, 5, z, (3/4)]//FunctionExpand

Possible Issues (1)

LameCPrime is not defined if is a negative integer:

Wolfram Language code: LameCPrime[ν, -3, z, m]

LameCPrime is not defined if is not an integer:

Wolfram Language code: LameCPrime[ν, 3 / 2, z, m]

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LameCPrime

Details

Examples

Basic Examples (3)

Scope (26)

Numerical Evaluation (5)

Specific Values (3)

Visualization (6)

Function Properties (2)

Differentiation (2)

Integration (3)

Series Expansions (3)

Function Representations (2)

Applications (1)

Properties & Relations (2)

Possible Issues (1)

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LameCPrime

Details

Examples

Basic Examples (3)

Scope (26)

Numerical Evaluation (5)

Specific Values (3)

Visualization (6)

Function Properties (2)

Differentiation (2)

Integration (3)

Series Expansions (3)

Function Representations (2)

Applications (1)

Properties & Relations (2)

Possible Issues (1)

See Also

Related Guides

History

Text

CMS

APA

BibTeX

BibLaTeX