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LameSPrime

LameSPrime[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.
LameSPrime belongs to the Lamé class of functions.
For certain special arguments, LameSPrime automatically evaluates to exact values.
LameSPrime can be evaluated to arbitrary numerical precision for an arbitrary complex argument.
LameSPrime automatically threads over lists.
LameSPrime[ν,j,z,0]=-j Cos[j(-z)].

Examples

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Basic Examples (3)

Evaluate numerically:

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

Plot the LameSPrime function for and :

Wolfram Language code:

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

Series expansion of LameSPrime at the origin:

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

Scope (26)

Numerical Evaluation (5)

Evaluate to high precision:

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

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

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

LameSPrime can take complex number parameters and argument:

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

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

Evaluate LameSPrime efficiently at high precision:

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

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

Lists and matrices:

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

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

Wolfram Language code:

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

Specific Values (3)

Value of LameSPrime when and :

Wolfram Language code: LameSPrime[ν, 1, z, 0]

Value of LameSPrime when and :

Wolfram Language code: LameSPrime[ν, 4, z, 0]

Some poles of LameSPrime:

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

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

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

Wolfram Language code: LameSPrime[5, 2, z, (1/2)]//FunctionExpand

Visualization (6)

Plot the first three even LameSPrime functions:

Wolfram Language code:

With[{ν = 3 / 2, m = 1 / 2}, 
	Plot[{LameSPrime[ν, 2, z, m], LameSPrime[ν, 4, z, m], LameSPrime[ν, 6, z, m]}, {z, -2EllipticK[m], 2EllipticK[m]}, Evaluated -> True]]

Plot the first three odd LameSPrime functions:

Wolfram Language code:

With[{ν = 3 / 2, m = 1 / 2}, 
	Plot[{LameSPrime[ν, 1, z, m], LameSPrime[ν, 3, z, m], LameSPrime[ν, 5, z, m]}, {z, -2EllipticK[m], 2EllipticK[m]}, Evaluated -> True]]

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

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

Plot LameSPrime as a function of its first parameter :

Wolfram Language code:

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

Plot LameSPrime as a function of and elliptic parameter :

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

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

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

Function Properties (2)

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

Wolfram Language code:

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

Wolfram Language code:

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

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

Wolfram Language code:

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

Wolfram Language code:

With[{ν = 7 / 2, m = 9 / 10, j = 3}, 
	Plot[LameSPrime[ν, j, z, m], {z, -4EllipticK[m], 4EllipticK[m]}, Epilog -> Point[{0, 0}]]]

Differentiation (2)

The -derivative of LameSPrime involves the LameS function:

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

Derivatives of LameSPrime for specific cases of parameters:

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

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

Integration (3)

Indefinite integral of LameSPrime is LameS:

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

Definite numerical integrals of LameSPrime:

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

More integrals with LameSPrime:

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

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

Series Expansions (3)

Series expansion of LameSPrime at the origin:

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

Coefficient of the second term of this expansion:

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

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

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

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

Wolfram Language code:

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

Series expansion for LameSPrime at any ordinary complex point:

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

Function Representations (2)

LameSPrime cannot be represented in terms of MeijerG:

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

TraditionalForm formatting:

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

Applications (1)

Use the LameSPrime function to calculate the derivatives of LameS:

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

Properties & Relations (2)

LameSPrime is an even function when is an positive even integer:

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

LameSPrime is an odd function when is an positive odd integer:

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

Use FunctionExpand to expand LameSPrime for integer values of and :

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

Possible Issues (1)

LameSPrime is not defined if is a negative integer:

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

LameSPrime is not defined if is not an integer:

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

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LameSPrime

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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LameSPrime

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