.. _sphx_glr_api_gallery_complex_analysis_holomorphic:

Holomorphic functions
---------------------

Euler's formula and the Cauchy-Riemann equations for complex differentiability.


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    <div class="sphx-glr-thumbcontainer" tooltip="Euler&#x27;s 1748 formula wraps the real line around the unit circle: e^{i\theta} is the point at angle \theta, and e^{i\pi} + 1 = 0. The domain coloring of e^z shows the consequence for the whole plane: the hue (phase) depends only on \operatorname{Im} z, so e^z repeats every 2\pi i.">

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  .. image:: /api/gallery/complex_analysis/holomorphic/images/thumb/sphx_glr_plot_01_euler_formula_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/holomorphic/plot_01_euler_formula`

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      <div class="sphx-glr-thumbnail-title">Euler's formula: e^{i theta} = cos theta + i sin theta</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="A function f = u + iv is complex differentiable exactly where its real and imaginary parts satisfy the Cauchy-Riemann equations. The residual \max(|u_x - v_y|, |u_y + v_x|) vanishes everywhere for z^2 and e^z, but not for \bar z, |z|^2, or \operatorname{Re} z. |z|^2 is the interesting case: it satisfies the equations only at z = 0.">

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  .. image:: /api/gallery/complex_analysis/holomorphic/images/thumb/sphx_glr_plot_02_cauchy_riemann_equations_thumb.png
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  :doc:`/api/gallery/complex_analysis/holomorphic/plot_02_cauchy_riemann_equations`

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      <div class="sphx-glr-thumbnail-title">The Cauchy-Riemann equations: u_x = v_y, u_y = -v_x</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Caspar Wessel (1797) and Jean-Robert Argand (1806) drew a + bi as the point (a, b). Addition is then vector addition, and multiplication by w = re^{i\varphi} rotates every point by \varphi and scales it by r. Multiplying by i is a quarter turn, so i^2 = -1 is a half turn.">

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  .. image:: /api/gallery/complex_analysis/holomorphic/images/thumb/sphx_glr_plot_03_wessel_argand_plane_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/holomorphic/plot_03_wessel_argand_plane`

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      <div class="sphx-glr-thumbnail-title">Wessel and Argand's complex plane: multiplication rotates and scales</div>
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.. toctree::
   :hidden:

   /api/gallery/complex_analysis/holomorphic/plot_01_euler_formula
   /api/gallery/complex_analysis/holomorphic/plot_02_cauchy_riemann_equations
   /api/gallery/complex_analysis/holomorphic/plot_03_wessel_argand_plane