HeunTPrime

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`HeunTPrime`

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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

open allclose all

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

Evaluate numerically:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-3kfjq

Out[1]=1

Plot HeunTPrime:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-ftt82q

Out[1]=1

Series expansion of HeunTPrime:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-z8evs7

Out[1]=1

Scope (22)Survey of the scope of standard use cases

Numerical Evaluation (8)

Evaluate to high precision:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-uqurky

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-lw9h0n

Out[1]=1

HeunTPrime can take one or more complex number parameters:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-64g5bd

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ywpcl7nzfzo-ft5oo1

Out[2]=2

HeunTPrime can take complex number arguments:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-hunut5

Out[1]=1

Finally, HeunTPrime can take all complex number input:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-56m4mo

Out[1]=1

Evaluate HeunTPrime efficiently at high precision:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-2c7v5i

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ywpcl7nzfzo-yaawua

Out[2]=2

Lists and matrices:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-22a9kq

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ywpcl7nzfzo-1knfqv

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0ywpcl7nzfzo-7yjugj

Out[3]=3

Compute the elementwise values of an array:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-thgd2

Out[1]=1

Or compute the matrix HeunTPrime function using MatrixFunction:

In[2]:=2

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https://wolfram.com/xid/0ywpcl7nzfzo-o5jpo

Out[2]=2

Specific Values (1)

Value of HeunTPrime at origin:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-nuboa

Out[1]=1

Visualization (5)

Plot the HeunTPrime function:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-n742f

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-35sv9o

Out[1]=1

Plot HeunTPrime as a function of its second parameter :

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-vhxvag

Out[1]=1

Plot HeunTPrime as a function of and :

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-mzgatp

In[2]:=2

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https://wolfram.com/xid/0ywpcl7nzfzo-8282mz

Out[2]=2

Plot the family of HeunTPrime functions for different values of the accessory parameter :

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-5yyyeh

In[2]:=2

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https://wolfram.com/xid/0ywpcl7nzfzo-dnzkk3

Out[2]=2

Differentiation (1)

The derivatives of HeunTPrime are calculated using the HeunT function:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-6eb2k6

Out[1]=1

Integration (3)

Integral of HeunTPrime gives back HeunT:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-ecaem6

Out[1]=1

Definite numerical integral of HeunTPrime:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-3rkya0

Out[1]=1

More integrals with HeunTPrime:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-gjk5w4

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ywpcl7nzfzo-q3siwd

Out[2]=2

Series Expansions (4)

Taylor expansion for HeunTPrime at origin:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-cq23zb

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-9rxgh1

Out[1]=1

Plots of the first three approximations for HeunTPrime around :

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-l68lro

In[2]:=2

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https://wolfram.com/xid/0ywpcl7nzfzo-j185j3

In[3]:=3

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https://wolfram.com/xid/0ywpcl7nzfzo-hrtnwe

Out[3]=3

Series expansion for HeunTPrime at any ordinary complex point:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-ukhgue

Out[1]=1

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

Use the HeunTPrime function to calculate the derivatives of HeunT:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-8yj5vx

Out[1]=1

Properties & Relations (4)Properties of the function, and connections to other functions

HeunTPrime is analytic at the origin:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-usyc66

Out[1]=1

HeunTPrime can be calculated at any finite complex :

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-txs34a

Out[1]=1

HeunTPrime is the derivative of HeunT:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-rvrd6q

Out[1]=1

Use FunctionExpand to expand HeunTPrime into simpler functions:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-h1b7ei

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ywpcl7nzfzo-ftheqb

Out[2]=2

Possible Issues (1)Common pitfalls and unexpected behavior

HeunTPrime calculations might take time for big arguments:

In[3]:=3

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https://wolfram.com/xid/0ywpcl7nzfzo-6fjqjs

Out[3]=3

Neat Examples (1)Surprising or curious use cases

The Schrödinger equation for the following infinite potential well can be solved in terms of HeunTPrime:

In[1]:=1

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https://wolfram.com/xid/0ywpcl7nzfzo-jjn6ko

Plot the potential:

In[2]:=2

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https://wolfram.com/xid/0ywpcl7nzfzo-rwdp9i

Out[2]=2

Construct the general solution of the Schrödinger equation:

In[3]:=3

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https://wolfram.com/xid/0ywpcl7nzfzo-kqhlks

In[6]:=6

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https://wolfram.com/xid/0ywpcl7nzfzo-piwc6c

Verify this solution by direct substitution:

In[7]:=7

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https://wolfram.com/xid/0ywpcl7nzfzo-c7bzyd

Out[7]=7

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