HeunTPrime

HeunTPrime[q,α,γ,δ,ϵ,z]

gives the -derivative of the HeunT function.

Details

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

Examples

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Basic Examples  (3)

Evaluate numerically:

Plot HeunTPrime:

Series expansion of HeunTPrime:

Scope  (21)

Numerical Evaluation  (7)

Evaluate to high precision:

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

HeunTPrime can take one or more complex number parameters:

HeunTPrime can take complex number arguments:

Finally, HeunTPrime can take all complex number input:

Evaluate HeunTPrime efficiently at high precision:

Lists and matrices:

Specific Values  (1)

Value of HeunTPrime at origin:

Visualization  (5)

Plot the HeunTPrime function:

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

Plot HeunTPrime as a function of its second parameter :

Plot HeunTPrime as a function of and :

Plot the family of HeunTPrime functions for different accessory parameter :

Differentiation  (1)

The derivatives of HeunTPrime are calculated using the HeunT function:

Integration  (3)

Integral of HeunTPrime gives back HeunT:

Definite numerical integral of HeunTPrime:

More integrals with HeunTPrime:

Series Expansions  (4)

Taylor expansion for HeunTPrime at origin:

Coefficient of the second term in the series expansion of HeunTPrime at :

Plots of the first three approximations for HeunTPrime around :

Series expansion for HeunTPrime at any ordinary complex point:

Applications  (1)

Use the HeunTPrime function to calculate the derivatives of HeunT:

Properties & Relations  (3)

HeunTPrime is analytic at the origin:

HeunTPrime can be calculated at any finite complex :

HeunTPrime is the derivative of HeunT:

Possible Issues  (1)

HeunTPrime diverges for big arguments:

Neat Examples  (1)

Schrödinger equation for the following infinite potential well is solved in HeunPrime functions:

Plot the potential:

The general solution of the Schrödinger equation is written in terms of HeunTPrime functions:

Verify this solution by direct substitution:

Introduced in 2020
 (12.1)