HeunGPrime

HeunGPrime[a,q,α,β,γ,δ,z]

gives the -derivative of the HeunG function.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • HeunGPrime belongs to the Heun class of functions.
  • For certain special arguments, HeunGPrime automatically evaluates to exact values.
  • HeunGPrime can be evaluated for arbitrary complex parameters.
  • HeunGPrime can be evaluated to arbitrary numerical precision.
  • HeunGPrime automatically threads over lists.

Examples

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Basic Examples  (3)

Evaluate numerically:

Plot the HeunGPrime function:

Series expansion of HeunGPrime:

Scope  (27)

Numerical Evaluation  (9)

Evaluate to high precision:

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

HeunGPrime can take one or more complex number parameters:

HeunGPrime can take complex number arguments:

Finally, HeunGPrime can take all complex number input:

Evaluate HeunGPrime efficiently at high precision:

Lists and matrices:

Evaluate HeunGPrime for points at branch cut to :

Evaluate HeunGPrime for points on a branch cut from to DirectedInfinity[a]:

Specific Values  (5)

Value of HeunGPrime at origin:

Value of HeunGPrime at regular singular point is indeterminate:

Value of HeunGPrime at regular singular point is indeterminate:

Values of HeunGPrime in "logarithmic" cases, i.e. for nonpositive integer , are not determined:

Value of HeunGPrime is not determined if :

Visualization  (5)

Plot the HeunGPrime function:

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

Plot HeunGPrime as a function of its third parameter :

Plot HeunGPrime as a function of and :

Plot the family of HeunGPrime functions for different accessory parameter :

Differentiation  (1)

The derivatives of HeunGPrime are calculated using the HeunG function:

Integration  (3)

Integral of HeunGPrime gives back HeunG:

Definite numerical integral of HeunGPrime:

More integrals with HeunGPrime:

Series Expansions  (4)

Taylor expansion for HeunGPrime at regular singular origin:

Coefficient of the first term in the series expansion of HeunGPrime at :

Plots of the first three approximations for HeunGPrime around :

Series expansion for HeunGPrime at any ordinary complex point:

Applications  (1)

Use the HeunGPrime function to calculate the derivatives of HeunG:

Properties & Relations  (3)

HeunGPrime is analytic at the origin:

is a singular point of the HeunGPrime function:

is a singular point of the HeunGPrime function:

Except for these two singular points, HeunGPrime can be calculated at any finite complex :

HeunGPrime is the derivative of HeunG:

Possible Issues  (2)

HeunGPrime cannot be evaluated if is a nonpositive integer (so-called logarithmic cases):

HeunGPrime is undefined when :

Introduced in 2020
 (12.1)