.. _sphx_glr_api_gallery_complex_analysis_residues:

Residues
--------

The residue theorem and the argument principle.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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

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    <div class="sphx-glr-thumbcontainer" tooltip="For f meromorphic inside \gamma, \oint_\gamma f = 2\pi i \sum_k n(\gamma, z_k)\operatorname{Res}(f, z_k). Growing a circle past the poles of f(z) = e^{z}/((z - 1)(z + 2)(z - 3i)) adds each residue in turn, and the same theorem evaluates the real integral \int_{-\infty}^\infty dx/(1 + x^4) = \pi/\sqrt 2.">

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  .. image:: /api/gallery/complex_analysis/residues/images/thumb/sphx_glr_plot_01_residue_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/residues/plot_01_residue_theorem`

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      <div class="sphx-glr-thumbnail-title">Cauchy's residue theorem: the contour integral is 2 pi i times the enclosed residues</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="\frac{1}{2\pi i}\oint_\gamma f&#x27;/f = N - P, the number of zeros minus poles inside \gamma. Equivalently, the image curve f(\gamma) winds N - P times around 0. For a quintic, counting roots inside disks of growing radius needs no root finder at all.">

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  .. image:: /api/gallery/complex_analysis/residues/images/thumb/sphx_glr_plot_02_argument_principle_thumb.png
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  :doc:`/api/gallery/complex_analysis/residues/plot_02_argument_principle`

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      <div class="sphx-glr-thumbnail-title">The argument principle: counting zeros minus poles by winding</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Gauss&#x27;s 1799 dissertation gave the first substantial proof that every polynomial of degree n has n complex roots. The modern topological proof counts windings: on a huge circle p(z) \approx z^n winds n times around 0, on a tiny circle about a non-root it winds 0 times, so as the circle grows the image curve must pass through 0 -- and the argument principle says it does so exactly n times.">

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  .. image:: /api/gallery/complex_analysis/residues/images/thumb/sphx_glr_plot_03_fundamental_theorem_of_algebra_thumb.png
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  :doc:`/api/gallery/complex_analysis/residues/plot_03_fundamental_theorem_of_algebra`

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      <div class="sphx-glr-thumbnail-title">The fundamental theorem of algebra, by winding numbers</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Pierre Alphonse Laurent (1843, after Weierstrass in 1841) expanded a function holomorphic in an annulus r &lt; |z - a| &lt; R as \sum_{n=-\infty}^{\infty} a_n (z-a)^n, with every coefficient a contour integral a_n = \frac{1}{2\pi i}\oint f(z)(z-a)^{-n-1}dz. For f(z) = 1/((z-1)(z-2)) the expansion depends on the annulus: in 1 &lt; |z| &lt; 2 it is a_n = -2^{-n-1} for n \ge 0 and a_n = -1 for n &lt; 0.">

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  .. image:: /api/gallery/complex_analysis/residues/images/thumb/sphx_glr_plot_04_laurent_series_thumb.png
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  :doc:`/api/gallery/complex_analysis/residues/plot_04_laurent_series`

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      <div class="sphx-glr-thumbnail-title">Laurent series: expansions with negative powers in an annulus</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Eugène Rouché (1862): if |g| &lt; |f| on a closed contour, then f and f + g have the same number of zeros inside it. Picture f(\gamma) as a person walking around a lamppost at 0 and g as the leash to a dog at f + g: a leash shorter than the distance to the post means the dog circles it the same number of times. Here z^5 + 3z + 1 has all five zeros in |z| &lt; 2 (dominated by z^5) and exactly one in |z| &lt; 1 (dominated by 3z).">

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  .. image:: /api/gallery/complex_analysis/residues/images/thumb/sphx_glr_plot_05_rouche_theorem_thumb.png
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  :doc:`/api/gallery/complex_analysis/residues/plot_05_rouche_theorem`

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      <div class="sphx-glr-thumbnail-title">Rouché's theorem: counting zeros by domination</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/complex_analysis/residues/plot_01_residue_theorem
   /api/gallery/complex_analysis/residues/plot_02_argument_principle
   /api/gallery/complex_analysis/residues/plot_03_fundamental_theorem_of_algebra
   /api/gallery/complex_analysis/residues/plot_04_laurent_series
   /api/gallery/complex_analysis/residues/plot_05_rouche_theorem