.. _sphx_glr_api_gallery_complex_analysis_contour_integrals:

Contour integrals
-----------------

Cauchy's integral theorem and Cauchy's integral formula.


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="For f holomorphic inside and on a closed contour \gamma, \oint_\gamma f(z)\,dz = 0, whatever the shape of \gamma. 1/z shows what happens when the hypothesis fails: its integral is 2\pi i times the number of times \gamma winds around the singularity at 0.">

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  .. image:: /api/gallery/complex_analysis/contour_integrals/images/thumb/sphx_glr_plot_01_cauchy_integral_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/contour_integrals/plot_01_cauchy_integral_theorem`

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      <div class="sphx-glr-thumbnail-title">Cauchy's integral theorem: the integral of a holomorphic function around a closed contour is zero</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="For f holomorphic inside a contour \gamma,">

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  .. image:: /api/gallery/complex_analysis/contour_integrals/images/thumb/sphx_glr_plot_02_cauchy_integral_formula_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/contour_integrals/plot_02_cauchy_integral_formula`

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      <div class="sphx-glr-thumbnail-title">Cauchy's integral formula: boundary values determine f and every derivative</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Cauchy&#x27;s integral formula on the circle |z| = R gives the estimate |f&#x27;(0)| \le M(R)/R, where M(R) is the maximum of |f| on the circle. If f is entire and bounded, letting R \to \infty forces f&#x27; \equiv 0. So every nonconstant entire function, such as \sin z, is unbounded. Applied to 1/p(z), the theorem proves the fundamental theorem of algebra.">

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  .. image:: /api/gallery/complex_analysis/contour_integrals/images/thumb/sphx_glr_plot_03_liouville_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/contour_integrals/plot_03_liouville_theorem`

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      <div class="sphx-glr-thumbnail-title">Liouville's theorem: a bounded entire function is constant</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/complex_analysis/contour_integrals/plot_01_cauchy_integral_theorem
   /api/gallery/complex_analysis/contour_integrals/plot_02_cauchy_integral_formula
   /api/gallery/complex_analysis/contour_integrals/plot_03_liouville_theorem