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Examples
========

This gallery walks through every public feature of ``mathematicskit.special_functions``:
the gamma/beta functions and Stirling's series, Bessel, Airy, and
Mathieu functions, elliptic functions, hypergeometric functions, the
error function and Fresnel integrals, the Riemann zeta function, the
Lambert W function, orthogonal polynomial families, and the discrete
Fourier transform (naive DFT vs. hand-rolled radix-2 FFT vs.
``numpy.fft``).

Each script in this gallery is self-contained and can be run directly with
``python examples/special_functions/<section>/<script>.py``.

Sections
--------

- **gamma_beta** -- the gamma and beta functions, and Stirling's
  approximation.
- **lambert_w** -- the Lambert W function and its two real branches.
- **elliptic** -- elliptic integrals, the arithmetic-geometric mean, and
  Jacobi elliptic functions.
- **hypergeometric** -- Gauss's and Kummer's hypergeometric functions.
- **error_functions** -- the error function and the Fresnel integrals.
- **bessel** -- Bessel functions of the first and second kind.
- **airy** -- the Airy functions Ai and Bi.
- **zeta** -- the Riemann zeta function and Euler's product.
- **mathieu** -- Mathieu functions and their characteristic values.
- **orthogonal_polynomials** -- Legendre, Chebyshev, Hermite, and
  Laguerre polynomials, with numerically verified orthogonality.
- **fourier_transform** -- naive DFT vs. radix-2 FFT vs. ``numpy.fft``.


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Airy functions
--------------

Solutions of ``y'' = xy``, oscillating on one side and exponential on the other.


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    <div class="sphx-glr-thumbcontainer" tooltip="Plots Ai and Bi across the turning point x = 0 where y&#x27;&#x27; = xy switches from oscillation to exponential behaviour, and checks the Wronskian Ai Bi&#x27; - Ai&#x27; Bi = 1/pi.">

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  .. image:: /api/gallery/special_functions/airy/images/thumb/sphx_glr_plot_01_airy_functions_thumb.png
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  :doc:`/api/gallery/special_functions/airy/plot_01_airy_functions`

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      <div class="sphx-glr-thumbnail-title">The Airy functions Ai and Bi</div>
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Bessel functions
------------------

Bessel functions of the first and second kind.


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    <div class="sphx-glr-thumbcontainer" tooltip="Plots J_0, J_1 (regular at the origin) and Y_0 (singular at the origin) over a range of x, and finds J_0&#x27;s first zero.">

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  .. image:: /api/gallery/special_functions/bessel/images/thumb/sphx_glr_plot_01_bessel_functions_thumb.png
    :alt:

  :doc:`/api/gallery/special_functions/bessel/plot_01_bessel_functions`

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      <div class="sphx-glr-thumbnail-title">Bessel functions of the first and second kind</div>
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Elliptic functions
------------------

Elliptic integrals, the arithmetic-geometric mean, and Jacobi elliptic functions.


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    <div class="sphx-glr-thumbcontainer" tooltip="Iterates the AGM by hand to show its quadratic convergence, then checks Gauss&#x27;s 1799 identity K(m) = pi / (2 AGM(1, sqrt(1 - m))) and uses E(m) for the perimeter of an ellipse.">

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  .. image:: /api/gallery/special_functions/elliptic/images/thumb/sphx_glr_plot_01_agm_elliptic_integrals_thumb.png
    :alt:

  :doc:`/api/gallery/special_functions/elliptic/plot_01_agm_elliptic_integrals`

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      <div class="sphx-glr-thumbnail-title">Gauss's arithmetic-geometric mean and the elliptic integral K</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Plots sn, cn, dn for m = 0.9, showing their real period 4K(m), and uses them to write the exact motion of a large-amplitude pendulum.">

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  .. image:: /api/gallery/special_functions/elliptic/images/thumb/sphx_glr_plot_02_jacobi_elliptic_functions_thumb.png
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  :doc:`/api/gallery/special_functions/elliptic/plot_02_jacobi_elliptic_functions`

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      <div class="sphx-glr-thumbnail-title">Jacobi elliptic functions and the pendulum</div>
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Error function and Fresnel integrals
------------------------------------

Gaussian integrals along the real axis and along the diagonal of the complex plane.


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    <div class="sphx-glr-thumbcontainer" tooltip="Draws the Euler-Cornu spiral (C(t), S(t)), which winds into the points (1/2, 1/2) and (-1/2, -1/2), and the knife-edge diffraction intensity that Fresnel computed from it.">

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  .. image:: /api/gallery/special_functions/error_functions/images/thumb/sphx_glr_plot_01_fresnel_cornu_spiral_thumb.png
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  :doc:`/api/gallery/special_functions/error_functions/plot_01_fresnel_cornu_spiral`

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      <div class="sphx-glr-thumbnail-title">Fresnel integrals and the Cornu spiral</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Recovers the 68-95-99.7 rule from erf, and shows why erfc is needed in the far tail, where 1 - erf(x) rounds to zero.">

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  .. image:: /api/gallery/special_functions/error_functions/images/thumb/sphx_glr_plot_02_error_function_thumb.png
    :alt:

  :doc:`/api/gallery/special_functions/error_functions/plot_02_error_function`

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      <div class="sphx-glr-thumbnail-title">The error function and the normal distribution</div>
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Discrete Fourier transform
----------------------------

Naive DFT vs. hand-rolled radix-2 FFT vs. ``numpy.fft``.


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    <div class="sphx-glr-thumbcontainer" tooltip="Cross-checks all three methods for correctness, then times them across a range of signal lengths to make the algorithmic speedup visible directly.">

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  .. image:: /api/gallery/special_functions/fourier_transform/images/thumb/sphx_glr_plot_01_naive_vs_radix2_vs_numpy_thumb.png
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  :doc:`/api/gallery/special_functions/fourier_transform/plot_01_naive_vs_radix2_vs_numpy`

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      <div class="sphx-glr-thumbnail-title">O(n^2) vs. O(n log n): naive DFT vs. radix-2 FFT vs. numpy.fft</div>
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Gamma and beta functions
---------------------------

The gamma and beta functions, and Stirling's approximation.


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    <div class="sphx-glr-thumbcontainer" tooltip="Confirms Gamma(n) = (n-1)! at integers, Gamma(1/2) = sqrt(pi), and the beta function&#x27;s relationship to gamma.">

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  .. image:: /api/gallery/special_functions/gamma_beta/images/thumb/sphx_glr_plot_01_gamma_extends_factorial_thumb.png
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  :doc:`/api/gallery/special_functions/gamma_beta/plot_01_gamma_extends_factorial`

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      <div class="sphx-glr-thumbnail-title">The gamma function extends the factorial</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Shows that sqrt(2 pi n) (n/e)^n approximates n! with relative error about 1/(12n), and that Stirling&#x27;s full series for log Gamma(x) is asymptotic: at small x, adding terms first helps and then hurts.">

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  .. image:: /api/gallery/special_functions/gamma_beta/images/thumb/sphx_glr_plot_02_stirling_formula_thumb.png
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  :doc:`/api/gallery/special_functions/gamma_beta/plot_02_stirling_formula`

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      <div class="sphx-glr-thumbnail-title">Stirling's formula and its asymptotic series</div>
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Hypergeometric functions
------------------------

Gauss's hypergeometric function and Kummer's confluent hypergeometric function.


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    <div class="sphx-glr-thumbcontainer" tooltip="Recovers log(1 + z), arcsin z, and a Chebyshev polynomial from 2F1 with special parameters, and checks Gauss&#x27;s summation theorem at z = 1.">

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  .. image:: /api/gallery/special_functions/hypergeometric/images/thumb/sphx_glr_plot_01_gauss_hypergeometric_thumb.png
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  :doc:`/api/gallery/special_functions/hypergeometric/plot_01_gauss_hypergeometric`

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      <div class="sphx-glr-thumbnail-title">Gauss's hypergeometric function contains the elementary functions</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Checks Kummer&#x27;s transformation M(a, b, z) = e^z M(b - a, b, -z) and shows 2F1(a, c; b; z/c) merging into 1F1(a; b; z) as c grows.">

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  .. image:: /api/gallery/special_functions/hypergeometric/images/thumb/sphx_glr_plot_02_kummer_confluent_thumb.png
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  :doc:`/api/gallery/special_functions/hypergeometric/plot_02_kummer_confluent`

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      <div class="sphx-glr-thumbnail-title">Kummer's confluent hypergeometric function</div>
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Lambert W function
------------------

The inverse of w e^w and its two real branches.


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    <div class="sphx-glr-thumbcontainer" tooltip="Plots W_0 and W_{-1}, the two real inverses of w e^w, which meet at the branch point z = -1/e, and uses W to solve x^x = 27.">

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  .. image:: /api/gallery/special_functions/lambert_w/images/thumb/sphx_glr_plot_01_lambert_w_branches_thumb.png
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  :doc:`/api/gallery/special_functions/lambert_w/plot_01_lambert_w_branches`

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      <div class="sphx-glr-thumbnail-title">The two real branches of the Lambert W function</div>
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Mathieu functions
-----------------

Periodic solutions of Mathieu's equation and their characteristic values.


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    <div class="sphx-glr-thumbcontainer" tooltip="Plots the first even Mathieu functions ce_m(x, q) and how the characteristic values a_m(q), b_m(q) split away from m^2 as q grows -- the band structure behind the Mathieu stability chart.">

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  .. image:: /api/gallery/special_functions/mathieu/images/thumb/sphx_glr_plot_01_mathieu_functions_thumb.png
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  :doc:`/api/gallery/special_functions/mathieu/plot_01_mathieu_functions`

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      <div class="sphx-glr-thumbnail-title">Mathieu functions and their characteristic values</div>
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Orthogonal polynomials
------------------------

Legendre and Chebyshev polynomials on [-1, 1], and Hermite and Laguerre
polynomials on infinite domains, with numerically verified orthogonality.


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    <div class="sphx-glr-thumbcontainer" tooltip="Plots the first Legendre polynomials P_n (weight 1) beside the first Chebyshev polynomials T_n (weight 1/sqrt(1 - x^2)), checks each family&#x27;s orthogonality, and confirms Chebyshev&#x27;s closed form T_n(cos t) = cos(nt), whose equal ripples between -1 and 1 are its minimal-oscillation property.">

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  .. image:: /api/gallery/special_functions/orthogonal_polynomials/images/thumb/sphx_glr_plot_01_legendre_chebyshev_thumb.png
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  :doc:`/api/gallery/special_functions/orthogonal_polynomials/plot_01_legendre_chebyshev`

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      <div class="sphx-glr-thumbnail-title">Legendre and Chebyshev polynomials on [-1, 1]</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Plots the first Hermite polynomials H_n (weight e^{-x^2} on the whole line) and Laguerre polynomials L_n (weight e^{-x} on the half-line), checks their orthogonality over infinite intervals, and draws the quantum harmonic oscillator states H_n(x) e^{-x^2/2} they describe.">

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  .. image:: /api/gallery/special_functions/orthogonal_polynomials/images/thumb/sphx_glr_plot_02_hermite_laguerre_thumb.png
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  :doc:`/api/gallery/special_functions/orthogonal_polynomials/plot_02_hermite_laguerre`

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      <div class="sphx-glr-thumbnail-title">Hermite and Laguerre polynomials on infinite domains</div>
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Riemann zeta function
---------------------

The zeta function, its analytic continuation, and Euler's product over primes.


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    <div class="sphx-glr-thumbcontainer" tooltip="Plots zeta(s) on the real line, including its analytic continuation to s &lt; 1 with trivial zeros at the negative even integers, and shows Euler&#x27;s product over primes converging to zeta(2) = pi^2/6.">

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  .. image:: /api/gallery/special_functions/zeta/images/thumb/sphx_glr_plot_01_riemann_zeta_thumb.png
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  :doc:`/api/gallery/special_functions/zeta/plot_01_riemann_zeta`

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      <div class="sphx-glr-thumbnail-title">The Riemann zeta function and Euler's product</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/special_functions/airy/index.rst
   /api/gallery/special_functions/bessel/index.rst
   /api/gallery/special_functions/elliptic/index.rst
   /api/gallery/special_functions/error_functions/index.rst
   /api/gallery/special_functions/fourier_transform/index.rst
   /api/gallery/special_functions/gamma_beta/index.rst
   /api/gallery/special_functions/hypergeometric/index.rst
   /api/gallery/special_functions/lambert_w/index.rst
   /api/gallery/special_functions/mathieu/index.rst
   /api/gallery/special_functions/orthogonal_polynomials/index.rst
   /api/gallery/special_functions/zeta/index.rst



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    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_