Complex Analysis

Table of Contents

Definitions

A multivalued function is a function which assigns to each point a set of values rather than a single value to each point, as a regular function would.

We define the open ball (or open disk since the complex plane is 2-dimensional) in the same way we do for the reals:

complex_analysis_b6a702e5f63b7d5d14ded5e33c27566833197d09.png

and the closed disk

complex_analysis_3727c0afb5c5c001c4a7dd009281ed26cba9d378.png

Open and closed sets in the complex plane then follow the same definition as for open and closed sets in the 2-dimension real case.

complex_analysis_e94dcc40f552ae52237c550aa588ad39c6b47ded.png

Functions

The *complex exponential function is the function complex_analysis_78571299104c46c514096e2f77f26f84b4998ef5.png defined by

complex_analysis_8a45516c4ba5b9d266e9df6fe9047ec00761a752.png

complex_analysis_e05cab8fabce58f4d74f3b7a234f8d621ae563b4.png

and

complex_analysis_772ed71a5e61f1bea57a9cf9db60a4d2d8a49fb6.png

Let complex_analysis_8a27571bd1409a3a4bfab6e0de2f41ad3d12d1de.png be non-zero. We define the multivalued function complex_analysis_2a64000ef310c81dbf1cc2e0e8f8ace28abb6ed2.png by

complex_analysis_8fe3f769b077698b4afdf0ff64063aab21191ad2.png

We call any element complex_analysis_fe480f547555026d210b55b5d4ef758235f32832.png of complex_analysis_2a64000ef310c81dbf1cc2e0e8f8ace28abb6ed2.png a logarithm of complex_analysis_9c15196dd07b1add486b8b54592e74bfe946ed95.png.

The principal branch of the logarithm function is the function complex_analysis_2c69087e1002c2a37b631d6a09b9676cc48fe218.png defined by

complex_analysis_7da3bfb67778c69ac7487d252bfa00a7939b62f5.png

for non-zero complex_analysis_8a27571bd1409a3a4bfab6e0de2f41ad3d12d1de.png, where complex_analysis_e4132500771fb9b225ec55839201c97fea7ed3c0.png is the principal value of the argument function.

Or more general

The branch cut of a complex function is defined to be the point in the principal branch where the function is no longer holomorphic.

Theorems

Holomorphic functions

Notation

  • complex_analysis_8a27571bd1409a3a4bfab6e0de2f41ad3d12d1de.png is almost always denoted complex_analysis_f1d820d252783cff9dffb8206c3555c3a06e8764.png
  • complex_analysis_63dfa6b68cf242ee64c39f27e9f5ef13c9fa0ff9.png
  • complex_analysis_4dbe6980c2b03faceb12b3792dc7861c6bb953a4.png, i.e. the unique complex_analysis_e7a86163f39fa21d4a2ed66946369cdeb900ef42.png in the range complex_analysis_19496ba4c6eb7980cdaa4858517e1376acae9b41.png
  • complex_analysis_a12be9db77b65a1e6ceb2dd092a03e2273f9e409.png is the complex plane without origin => can be equipped with multiplication operation to form an (Abelian) group
  • complex-valued function of a real variable means complex_analysis_ac7296e0e718c33ca1feb8f63852c44b2408e12b.png, i.e. input is real and we can decompose the output into a real part and an imaginary part
  • complex_analysis_f1f02b892dde2c81a7f77c7458c5e0370c97a0d5.png
  • complex_analysis_ab91f46db13bbe19acfc7ed74e35f876c0c3dea8.png

Differential Geometric view on arg function

Let

  • complex_analysis_78801776d53163d3dc9f1b130fb28b217233485a.png
  • complex_analysis_3e2fa5f05445d5631297172cbf268e6749e16108.png defined by complex_analysis_c987dbdeb172befb4308f1a5a8fe4f862610a649.png, which is then surjective and continuous
  • complex_analysis_50e6860ebe26cbfb881b13877422fd086fb4e633.png

Apparently, we call a complex manifold such as the one above a Riemann surface, and I honestly believe this is something I ought to return to once I have a better understanding of what complex manifolds are.

Observe then that this defines a fibre bundle, since for any two points complex_analysis_9e7e973118507d0ab7deda86371fabe2fc01a8be.png we have the fibres be isomorphic, i.e. complex_analysis_7fb649c0690e4b4310a6fd0d223ccb94c7df638d.png.

complex_analysis_bc4d8c2b4155ba3e3f727e5dc3041a210b747a1a.png is the plane defined by Cartesian product complex_analysis_ed1e63020ddcbdc9db25312774bf86317d11260c.png since complex_analysis_57ff2e6138d68a0116de4663cf3eec83d3c21ee7.png and complex_analysis_69eed04a103d5acd55bed321c31316de896a9534.png. Thus, complex_analysis_bc4d8c2b4155ba3e3f727e5dc3041a210b747a1a.png is in fact a topological manifold (probably equipped with the standard topology, but I haven't checked). Further we observe that complex_analysis_50e6860ebe26cbfb881b13877422fd086fb4e633.png is also a topological manifold. Thus, we our fibre bundle as claimed earlier.

Complex-valued functions

Suppose we have a complex-valued function of a real value

complex_analysis_aaeaac13b18bdfc8cc62d1d6053c4f9a62f67158.png

We then say that complex_analysis_be61dd3b009faae59aa34a7eeb9c1ee5d1bb1bde.png is continuous at complex_analysis_01df804acf60553200444b51ed91c83a9d8283d4.png if and only if both complex_analysis_0cd0a364dae52a06daa9dfd6c7f4b6a06a816117.png and complex_analysis_17871a95eb54b4106f807e1ca07127dbfaec65b8.png are continuous at complex_analysis_01df804acf60553200444b51ed91c83a9d8283d4.png.

If complex_analysis_be61dd3b009faae59aa34a7eeb9c1ee5d1bb1bde.png is instead a complex-valued function of a complex variable, let complex_analysis_6c1d964cf1d1448007ed57a3bf703f72dacca565.png, then

complex_analysis_fa7a052dcb08d8157ddebc946bdad7f712b7a127.png

We then say that complex_analysis_31e6c70555a855c0badc68901951b65b1a447a6f.png is continuous at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png if and only if for any complex_analysis_95253ea0c2082f08ac36eed736be50b9577033e8.png there exists complex_analysis_29d257aee0166cee8e1f0fcdd63eeebd87e090d7.png s.t.

complex_analysis_77ed38ed2705c257b1a96037e0af0fa025ce4c41.png

where complex_analysis_a5147769370557bd0de62b61ea2eb7e30e954b44.png and complex_analysis_75a1c35c77475fbf9ec8db8d6c8a6e29a4bcced6.png are real valued functions, and this is thus equivalent of complex_analysis_de4b3256c9441a9fb95ba92da7c59c7d1f70d1c7.png and complex_analysis_ebac26f7eba81eb516f423f028596d8e6018aa7a.png being continuous at the point complex_analysis_a0e990391dead6fbbc362b8f21c4307999efcf30.png.

Complex differentiability and holomorphicity

A function complex_analysis_443078545102001f9f9f6cddfd4b98f32601f71b.png in a neighborhood complex_analysis_5760e806c8eb938d6a2563d076feec6ab9f9cafc.png of complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png is differentiable (everywhere in complex_analysis_5760e806c8eb938d6a2563d076feec6ab9f9cafc.png) if and only if its real and imaginary parts complex_analysis_de4b3256c9441a9fb95ba92da7c59c7d1f70d1c7.png and complex_analysis_ebac26f7eba81eb516f423f028596d8e6018aa7a.png are continuously differentiable and obey the Cauchy-Riemann equations (everywhere in complex_analysis_5760e806c8eb938d6a2563d076feec6ab9f9cafc.png):

complex_analysis_8a4f6eecd566b5ca9fbebc7db955f35fe71305ff.png

This can be seen by considering the "normal" definition of differentiability at a point, and observing the definition changes depending on which direction we approach the point from (vertical or horizontal direction).

Important: it's crucial that the function be differentiable in a neighborhood complex_analysis_5760e806c8eb938d6a2563d076feec6ab9f9cafc.png of complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png not just at the point complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png!

We say that the complex-valued function complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is holomorphic at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png if it is differentiable everywhere in some neighbourhood complex_analysis_5760e806c8eb938d6a2563d076feec6ab9f9cafc.png of complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png.

We say that a function is entire if it is holomorphic in the whole complex plane.

Let complex_analysis_8496204ae32ef580844a6fa3ef4ffe0dd0a6f4de.png.

Then complex_analysis_3807b8495f01004f1d88370e2f4fad2f5032db06.png is harmonic if it satisfies the Laplace equation:

complex_analysis_7a5cd18555869d3ae57e7ae3e50dfea4c9913437.png

Let complex_analysis_4e36253a9de2af21dc489193ecb95b97ccccc6e0.png be open, and let complex_analysis_df4ed9b640263d4030989b62ce00967263b2fe84.png be harmonic.

We say a harmonic function complex_analysis_73c98061309c546322b6bb044d6e3c34dabe6be8.png is the harmonic conjugate of complex_analysis_de4b3256c9441a9fb95ba92da7c59c7d1f70d1c7.png if the complex-valued function

complex_analysis_b6f04d74028ff6a366b97039a43fa2229cecd45f.png

is holomorpic on complex_analysis_5760e806c8eb938d6a2563d076feec6ab9f9cafc.png.

Solving differential equations

This techinque can be used to solve differential equations on "ugly" domains.

Consider the following:

Polynomials and rational functions

Complex powers

Let complex_analysis_f5b492fd8bc04bc447c0587182466f68a479bf00.png with complex_analysis_be65c7839a969ee2e184afe84de0c0bc5c6c0c43.png. Then we define complex_analysis_6e9d20113f7ef024a71fc3edcc10e77417d06393.png th power of complex_analysis_9c15196dd07b1add486b8b54592e74bfe946ed95.png by

complex_analysis_3e4899a8c549a23181d8dbfd1b991d37ef74aee3.png

Unless stated otherwise, a complex power is defined in the principal branch.

Let complex_analysis_f5b492fd8bc04bc447c0587182466f68a479bf00.png with complex_analysis_be65c7839a969ee2e184afe84de0c0bc5c6c0c43.png. Then

  1. if complex_analysis_f6cc5909af95816c55fb57396661410cb0085a43.png there is exactly one value of complex_analysis_9c43bebac3f949b98a3763974c1d44efa2572f1d.png
  2. if complex_analysis_0e1c16255d889dbbcf3adb4c3d44ccdfccd60375.png where complex_analysis_a0b5d9cdf21d921c9f474ba47ff5ea047bbc7ec2.png are coprime, with complex_analysis_a7ab205986b98d56a9af2bbbb776f9a313a5cd23.png there are exactly complex_analysis_ab437e1f9b3376761b155efe111c9860607c4b86.png values of complex_analysis_dfeeec91b4a0cd07519e58141119b43531936e02.png
  3. if complex_analysis_a68b4dd98cb2875f380165bc85cbce8b02fe0948.png or complex_analysis_473809752ae0380890e20b6d6906e23428d2fd22.png, there are infinitely manu values of complex_analysis_9c43bebac3f949b98a3763974c1d44efa2572f1d.png

Let complex_analysis_81163e4aa9a7bd934c4fa36119b2296bfa5b4c1d.png. Then the complex_analysis_ab437e1f9b3376761b155efe111c9860607c4b86.png values

complex_analysis_935499b6376dd5be743596c913e323f46ba080b2.png

where

complex_analysis_ee994bab772c9735ea9f3e4fa3e9bc7a02c809d4.png

are the complex_analysis_ab437e1f9b3376761b155efe111c9860607c4b86.png roots of unity.

Graphing complex functions

  • Hard to visualize as 2D + 2D becomes 4D
  • Good idea to treat each of the different 3D plots separately

Let complex_analysis_b4afb7b2eee809aae29f62450a15fecd0c356648.png be open, and let complex_analysis_443078545102001f9f9f6cddfd4b98f32601f71b.png.

We say complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is conformal if complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png preserves angles: i.e. if the angle between the images under complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png of two straight lines in complex_analysis_5760e806c8eb938d6a2563d076feec6ab9f9cafc.png are equal to the angle between the two straight lines themselves.

By applying this definition to tangents of differentiable curves, more generally we can say the same about the angles between curves at certain points.

We're saying the push-forward of complex_analysis_9f093056c0fe301e7787fa3c1ebce1a9e96d5b30.png, is angle-preserving.

Therefore, any diffeomorphism between complex_analysis_5760e806c8eb938d6a2563d076feec6ab9f9cafc.png and complex_analysis_dac8840afafdef6e6a95f235be1e040e7f5a94e5.png is angle-preserving.

Consider complex_analysis_365e9f22017f08e70a3f14b85364e1c436db1e84.png. For complex_analysis_be65c7839a969ee2e184afe84de0c0bc5c6c0c43.png, complex_analysis_54047a3989550ef001ee16bbc1f578811442b4b7.png. Then we consider the surface

complex_analysis_aaab3462937aabfebef2c5af9b3d601550aade5b.png

and the function

complex_analysis_e867edd9c925257d9241fd74b7d5f0d3f33ffcb3.png defined by complex_analysis_f1088c056192741f25208fd58210127b83e46a77.png.

Let complex_analysis_b4afb7b2eee809aae29f62450a15fecd0c356648.png be open, and let complex_analysis_443078545102001f9f9f6cddfd4b98f32601f71b.png be holomorphic.

Then complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png preserves angles at every complex_analysis_3e61f3c1ae92256436410aa5f5e37d4aa1228e48.png where complex_analysis_84359b910929fbefee3032659a47a03d00e3d713.png.

This theorem is useful for establishing the image of a function (if it's a polytope (?)) since we can simply compute the mapped values at the edges (i.e. curves from vertex to vertex) and, knowing that the angles are preserved, immediately know how the edges between the mapped vertices look.

Möbius transformations

A Möbius transformation is a function of the form

complex_analysis_332128063265babdfcb154a0aec35d0118934b24.png

where complex_analysis_ea8d960c4503260261c5fd196da20fb559ee17cb.png are such that complex_analysis_4dee1cf1f4c47d043fa7f65e9900423bc15c2129.png.

Observe that complex_analysis_3cde5172d50d013ad7e2a092d995a909e9739757.png is not defined on the entirety of complex_analysis_dac8840afafdef6e6a95f235be1e040e7f5a94e5.png, which leads us to defining the extended complex plane.

We'll often consider the case complex_analysis_9eeedba94db42a9187edc7b60bb15525c06e72ff.png, if nothing else is specified.

That is, a Mobius transformation is defined on complex_analysis_21202ebd38431eb28de9a260ae3d938d34e9c7ef.png.

The extended complex plane is the set complex_analysis_3dc298e644dd995b4ec11fa0cd6017538eb59f31.png, where complex_analysis_30dd1bf5261c6231fab304afac9382dac65bc232.png is just some object complex_analysis_ee02056c09e17e53bc6f3ef5e1d2d7c03165853a.png.

We extend the usual arithmetic operations in the following way: for complex_analysis_8e6d4f25f0da25852fdfd5c02551852edf670fab.png and non-zero complex_analysis_02712cdbafb5bd35a2565c5d5da847d1650017e2.png,

complex_analysis_0278b9fdf2bd44a83b474193b1d47c9bb65dc888.png

Consider the coordinates complex_analysis_139f5a22f87ee5229c75a67eb24bd95a3e7e3feb.png describing complex_analysis_593ec27432c318ac6fbe2b8a7b3ffadcfdb140ff.png. We identify the complex plane with the plane defined by complex_analysis_042e56f3905e097269bd4e50fdda2ef0812a22cb.png, and a complex number complex_analysis_a63460ec6056998844afb523acdd134473499083.png with the point complex_analysis_8ec7239929fd7a8f4a266edae616657b7d93c3f4.png.

The Riemann sphere is the unit sphere complex_analysis_048a449f5828e2a289657408a9e03a9bd7a3a444.png in complex_analysis_593ec27432c318ac6fbe2b8a7b3ffadcfdb140ff.png defined by

complex_analysis_af91f06a29fd4d5c0860140ea54248fc4c56230a.png

and we consider the "north pole" to be the point complex_analysis_856dae543ea9d30fbed89c386856f9a8d1420f62.png.

The Riemann sphere therefore has two charts:

  • For all points in the complex plane, the chart is the identity map from the sphere (with complex_analysis_30dd1bf5261c6231fab304afac9382dac65bc232.png removed) to the complex plane.
  • For complex_analysis_30dd1bf5261c6231fab304afac9382dac65bc232.png, the chart neighborhood is the sphere (with the origin removed), and the chart is given by sending complex_analysis_30dd1bf5261c6231fab304afac9382dac65bc232.png to complex_analysis_d83ba0b34682ac87f8d84c8310f6bbd28d1fe65b.png and all other points complex_analysis_9c15196dd07b1add486b8b54592e74bfe946ed95.png to complex_analysis_b1eaf73631f47b3928057ddfe792212fe166818a.png.

Let complex_analysis_65c4c8e8ff84d812bfce3c9b96ef9f0535aa4643.png such that the three points complex_analysis_e10e2b430f95617381cdd6d6b52aed29fb971dff.png, complex_analysis_9c15196dd07b1add486b8b54592e74bfe946ed95.png, and complex_analysis_6475e318f51222f065729419b50ebb9ade4c3dd4.png are colinear.

It is clear that

complex_analysis_15de691042e8a7246b427dd3dd0946144f12c717.png

thus we define also complex_analysis_e6a3d9408d9cc2ad4feff2a70e42d49c01b9a342.png and thereby consider complex_analysis_2a275ba57b65332c8eb71ed5f6713abb5db0e66b.png as being defined on the extended complex plane, i.e. complex_analysis_e0fc29a2cfc3f6f08aec7b869f4aa204c8cf98fd.png.

The map complex_analysis_e0fc29a2cfc3f6f08aec7b869f4aa204c8cf98fd.png is evidently bijective, so it has an inverse complex_analysis_3263f23d3b5d55157912dd38d45ef604337bb03b.png. This function complex_analysis_d952e5ba43a4e985c72e92c3a15ac9a90f5f272a.png is the sterographic projection.

Better description:

The unit sphere in complex_analysis_593ec27432c318ac6fbe2b8a7b3ffadcfdb140ff.png is the set of pints complex_analysis_5499c2fd0d099625850bc87e12348c0638af92ab.png such that complex_analysis_a1185c993e80def5155c0c01a6fc1ff70a43a55e.png.

Let complex_analysis_34fbe262a44df903fe37d2fbbf6711bdd09de32d.png be the "north pole", and let complex_analysis_048a449f5828e2a289657408a9e03a9bd7a3a444.png be the rest of the sphere.

The plane complex_analysis_212dafde8d2b8be8c0a2822fd38dc60f3bd918bb.png (xy-plane) runs through the center of the sphere; the "equator" is the intersection of the sphere with this plane.

For any point complex_analysis_fd9fe5003b5dfa76cd57dccf213781e787a553d0.png on complex_analysis_048a449f5828e2a289657408a9e03a9bd7a3a444.png, there is a unique line through complex_analysis_e10e2b430f95617381cdd6d6b52aed29fb971dff.png and complex_analysis_fd9fe5003b5dfa76cd57dccf213781e787a553d0.png, and this line intersects the plane complex_analysis_212dafde8d2b8be8c0a2822fd38dc60f3bd918bb.png in exactly one point complex_analysis_3aa9247c25694aee69b7c9ec8ee4bc55c4896620.png. We define the stereographic projection of complex_analysis_fd9fe5003b5dfa76cd57dccf213781e787a553d0.png to be this point complex_analysis_3aa9247c25694aee69b7c9ec8ee4bc55c4896620.png in the plane.

complex_analysis_20b39d952e76acca81bb3849a170186185d7b746.png

and

complex_analysis_5f644bc8db368de937ad6e3e0fe3bdc1da58eeb8.png

6zeHc.jpg

Stereographic projection maps a circle to either a circle or a straight line (a "circline").

Makes some people say that straight lines are circles of infinite radius.

  1. A translation is a Möbius transformation of the form

    complex_analysis_0d20a63ee224ab3002b81d38deea08bd7e45bbc8.png

    which corresponds to the matrix

    complex_analysis_ef25eae651e92021bbc0470aa50389137b2076c2.png

  2. A rotation is a Möbius transformation of the form

    complex_analysis_ed209f84e53e9849a9fecb7762c814c207f78bc1.png

    so that complex_analysis_d60a2b2d080cdc468425d9ea9bf8d3f1953ee3c7.png for some complex_analysis_e7a86163f39fa21d4a2ed66946369cdeb900ef42.png, which corresponds to the matrix

    complex_analysis_e68dbf98856e8cd3e0171b733a0f1bc33a4c8f41.png

  3. A dilation is a Möbius transformation of the form

    complex_analysis_296d484f04675a0c2ab664b19e458522352a0bf2.png

    which corresponds to the matrix

    complex_analysis_7a9d6dee8b7145f49f50273208a11b1cc88940d0.png

  4. An inversion is a Möbius transformation of the form

    complex_analysis_c03a92a50201f158ca3efaaa82284e86c6786b28.png

    which corresponds to the matrix

    complex_analysis_e0a84c8412f57c6dcef9a6f88160c69e13edac60.png

We say that a Möbius transformation complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png fixes the point of infinity if complex_analysis_a0fdac27b5d23e729b257deb51f76c49bbd5a044.png.

Translations, rotatons, and dilations fix the point at infinity, while inversions do not.

Let complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be a Möbius transformation. Then complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is a composition of a finite number of translations, rotations, diluations, and, if and only if complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png does not fix the point at infinity, one inversion.

Möbius transformations map circlines to circlines.

Let complex_analysis_9dcb9de155d0ea85bcc6dbd14b4c6236ccebd014.png be three distinct points. Then there exists a unqiue Möbius transformation complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png such that

complex_analysis_f8114c357b7bc44e8fa935a513c76d474860b4b5.png

This is useful because we by simply knowing how the Möbius transformation maps the three different points, we can tell what it does to circles or lines.

Let complex_analysis_d427bba1e4ddbd9f68ad6eda86b572e5723d7f8e.png be distinct points.

The cross-ratio complex_analysis_ba1b274f861ae7ab9b6ff56667951dce04e9ddb1.png of the four points is the image of complex_analysis_8695baa5ba4d5b19604a9593a10a84def7b6564b.png under the Möbius transformation which sends complex_analysis_3e6d388ff5c525021be3cf14278ca883a796e418.png to complex_analysis_d13f3d7aef6c8c36ff460b72d2d06349a057c5c0.png.

Complex integration

Complex integrals

Let complex_analysis_1000324b82811b063e2bc91bf6447539f5483ac8.png be a interval, and complex_analysis_1d5d9e03c9e3d5dfc15d8acf60b307ef428fa670.png of the form

complex_analysis_b6f04d74028ff6a366b97039a43fa2229cecd45f.png

Then complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is integrable if its real and imaginary parts complex_analysis_7abea6c80fa3b363f461473fed1e5882d7ab0a33.png are integrable in the usual (real) sense, and we define the integral of complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png by

complex_analysis_09b983ab75fc08f970022a5b4bf72ba290554005.png

It will usually suffice to observe that continuous functions are integrable.

Example

complex_analysis_7e96de175cba939e395d8350691a254c05de6613.png

where complex_analysis_86ca21612a2d49233cf96085b37403bab3899ad6.png denotes the arc-length of complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png.

  • complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png is by def. closed and bounded, i.e. a compact set
  • complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is continuous

Hence complex_analysis_657112ef71fc15ebbbbbe3f94332fe97bfa22dfb.png is in fact bounded i.e. finite.

Contour integrals

Let complex_analysis_ff596ba638f52d81b40556d17c7b3e54705cfaf6.png be distinct.

Then a (parametrized) curve complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png connecting complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png and complex_analysis_8695baa5ba4d5b19604a9593a10a84def7b6564b.png is a continuous function

complex_analysis_8814c4439ebfbfbb7d991b77aa908df5ecca8bc0.png

Writing complex_analysis_0de2bf5310ebca2c16c0e0159192d6b7d1e02984.png and complex_analysis_4b755811a22ef67bba7e57ff391f2be4816df8be.png, we decompose complex_analysis_9c15196dd07b1add486b8b54592e74bfe946ed95.png into a real and imaginary parts for continuous real functions complex_analysis_2d90437e8fc89ef50863f60032a42744c7144acc.png, so

complex_analysis_86bb4205e1e2396e2fab94ef669e48bc6443265f.png

We say the curve complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png is regular if complex_analysis_339c704fc9bf0c2f6780a0a0e68b51e128bb7f74.png is continuously differentiable and complex_analysis_a271d1043108d247621f680d25ae656697cc3d23.png forall complex_analysis_533e14c29cf6e9eb8c5500e1f00ef278cef9b9fa.png.

A curve complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png from complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png to complex_analysis_8695baa5ba4d5b19604a9593a10a84def7b6564b.png in complex_analysis_dac8840afafdef6e6a95f235be1e040e7f5a94e5.png is a contour if it is a finite union of regualr curves, which together joint complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png with complex_analysis_8695baa5ba4d5b19604a9593a10a84def7b6564b.png, i.e. there exists

complex_analysis_13c333f5942eb33dea0380e4b26f200fe3c6bb15.png

such that

complex_analysis_a26cee27bbd3db249d1b68996a88c67c54f73cd2.png

such that complex_analysis_dd606cbabd0c695a73aa5f895a839a8f5cfb4b56.png is continuous.

For a continuous function complex_analysis_1d1ac38399684564b6111c26821caa91b3a301de.png, we define the contour integral of complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png along complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png by

complex_analysis_2e3f18b79bf196fb7e339e2e0f13d1aca1191d04.png

Let complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png be a curve in complex_analysis_dac8840afafdef6e6a95f235be1e040e7f5a94e5.png, and let complex_analysis_1d1ac38399684564b6111c26821caa91b3a301de.png be continuous. Then

complex_analysis_3c725f63d5d93cca859bcc886fd3a910fc1b2d13.png

complex_analysis_7e96de175cba939e395d8350691a254c05de6613.png

Let complex_analysis_29c2c2b8c474a8661a5d8718fe0fb256c8e0c2a9.png. We will say that complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png is a domain if complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png is open and every two points in complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png can be connected by a contour which lies wholly on complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png.

Let complex_analysis_dc7e39d8e1b97836f49daa4a56d4b47d7855cea7.png be a domain, and complex_analysis_97067d06e043a9083fb622b87bcd20f3604964a6.png be continous. Then the following are equivalent:

  1. complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png has an antiderviative complex_analysis_e08421ac16591ee60adecc1e14a359550f81a1e2.png on complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png
  2. complex_analysis_718689ed7a744d2d097f7e14181d1cf7dfb2ef19.png for all closed contours complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png in complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png
  3. all contour integrals complex_analysis_3ef41d1635d004613db09621beb29eab836af8d2.png are independent of path complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png, and depend only on the endpoints.

Cauchy's Integral Theorem

Let complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png be a contour in complex_analysis_dac8840afafdef6e6a95f235be1e040e7f5a94e5.png.

Then complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png is simple if it has no self-intersections, except possible at the endpoints, i.e. complex_analysis_9364acde6eb8b2966f7ca954c1a684d281737213.png for all distinct complex_analysis_5b64b510082b0afffa4e3678c6ca3a77cde0d88d.png, unless complex_analysis_f42d9358b8f097e976f41c1e8078da977da2ee95.png and complex_analysis_cfeee0d4790ac7fa8a780a36a09e1cf0dffca26e.png and complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png is a closed contour.

A loop is a simple, closed contour.

Let complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png be a loop in complex_analysis_dac8840afafdef6e6a95f235be1e040e7f5a94e5.png. Then complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png defines two regions in the complex_analysis_dac8840afafdef6e6a95f235be1e040e7f5a94e5.png, with complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png as their common boundary:

  • a boundary domain, the interior of complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png
  • an unbounded domain, the exterior of complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png

Let complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png be a loop in complex_analysis_dac8840afafdef6e6a95f235be1e040e7f5a94e5.png. We say complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png is positively-oriented if as we move along the curve in the direction of parametrization, the interior is on the LHS.

Otherwise state, all loops should be assumed to be positively-oriented.

complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png is said to be simply connected if the interior of every loop on complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png lies wholly in complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png.

Let complex_analysis_5fbda07c45a608d0477cc50191ca91954f0ba2ca.png and complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png be a loop in complex_analysis_dac8840afafdef6e6a95f235be1e040e7f5a94e5.png which does not pass through complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png. Then

complex_analysis_39912d2be3012fab953066a8b24cb2edd5651476.png

Since complex_analysis_bc319af221b7dc20b5d4e24ea87fbf52c4395a37.png, we clearly have two cases:

  1. complex_analysis_338850f7e1cf3f1702f8138d97d1ca837e1af313.png: integral is zero due to Cauchy Integral theorem
  2. complex_analysis_a8adac587cf5eabaef89a34a32848911914313f1.png: Consider the following figure: integral-about-closed-loop.png Since complex_analysis_a8adac587cf5eabaef89a34a32848911914313f1.png this figure basically "encapsulates" all possible loops which have complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png outside!

Let complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png be a loop, and complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be holomorphic inside and on complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png.

Then,

complex_analysis_f9c17cbac147485f8073714223b688eb0442d027.png

We break the integrand complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png, aswell as the differential complex_analysis_8345e5be5e96428e56e48eacb8b9eb7ed99a38f4.png into their real and imaginary components:

complex_analysis_34cdb0865c029e9acac487e805090dc570544013.png

In this case we have

complex_analysis_4a0946ce14751f50638419d37e6eea8db016e648.png

By Green's Theorem, we may then replace the integrals around the closed contour complex_analysis_339c704fc9bf0c2f6780a0a0e68b51e128bb7f74.png with an area integral throughout the domain complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png that is enclosed by complex_analysis_339c704fc9bf0c2f6780a0a0e68b51e128bb7f74.png as follows:

complex_analysis_874871cdac3d80045b837997fa75e8c67a9b5f40.png

and for imaginary part,

complex_analysis_45e6ceda27474a664e8b8aa97fefdc0ba6efcfab.png

However, being the real and imaginary parts of a function holomorphin in the domain complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png, complex_analysis_de4b3256c9441a9fb95ba92da7c59c7d1f70d1c7.png and complex_analysis_ebac26f7eba81eb516f423f028596d8e6018aa7a.png must satisfy the Cauchy-Riemann equations there:

complex_analysis_c40c2a9cc0b8098845be294a30bf3391d2a75b78.png

We therefore find that both integrads (and hence their integrals) are zero:

complex_analysis_67991d40a13a27b4f60f9d5879d784e9ba8e356b.png

Which gives us the result

complex_analysis_04d4de97583c3a4f9a378abdb5e95391da4fdfe8.png

as wanted.

Let complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png be a loop, complex_analysis_338850f7e1cf3f1702f8138d97d1ca837e1af313.png, and complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be holomorphic inside and on complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png. Then

complex_analysis_513a46aaccedf5f24d19ee876f00f0fc705f95f3.png

Let complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png be a loop, complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be holomorphic inside and on complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png, and complex_analysis_9c15196dd07b1add486b8b54592e74bfe946ed95.png lie inside complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png.

Then complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is infinitively differentiable at complex_analysis_9c15196dd07b1add486b8b54592e74bfe946ed95.png and, for all positive integers complex_analysis_f07b932b12c91bca424fd428d383495413b5b6a9.png,

complex_analysis_12ccecf8ffa28576cad5aba78ddbc65c3d38bbec.png

Let complex_analysis_dc7e39d8e1b97836f49daa4a56d4b47d7855cea7.png be a domain, complex_analysis_4ef3444614b45065eb24938a20e418d3d6213c78.png be continuous, and suppose that

complex_analysis_f9c17cbac147485f8073714223b688eb0442d027.png

for all loops complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png inside complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png.

Then complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is holomorphic.

Liouville's Theorem and its applications

Let complex_analysis_dc7e39d8e1b97836f49daa4a56d4b47d7855cea7.png be a domain, complex_analysis_c4f806c2ed58a3035e03bf180e0ca340e99a3aa9.png and complex_analysis_8ae53a3d04627802bc79bbde92987929abdb717e.png be such that complex_analysis_925c6f2c6ca02aedfd675a001c43667cc20ca333.png, complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be holomorphic on complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png, and complex_analysis_06c4a09d92b851bc0cf8a519f85459438f5959c4.png be such that complex_analysis_8b5f15e5d4e927bba73eff57c28ef4abb1948c6a.png for all complex_analysis_e87fb4ed76655c50da322f977d89a150edb9d855.png.

Then for all complex_analysis_c90914e0c097c39e2693a57eb49743ba7b3573ad.png, we have that

complex_analysis_a50a15b917e0d3f44830aa733d5a767ebffb4bfe.png

Let complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be holomorphic on complex_analysis_dac8840afafdef6e6a95f235be1e040e7f5a94e5.png and be bounded, i.e. satisfying

complex_analysis_3a595ec0a96fc8ef29a25b9d8f84234222085204.png

then complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is constant.

Maximum Modulus Principle

complex_analysis_8306532ce8294dd4701f9a1a5f912816f4d009f1.png and complex_analysis_8ae53a3d04627802bc79bbde92987929abdb717e.png be such that the closed disc complex_analysis_3bcb4f69964763b5ad610e4726d16cc0d7bf37a6.png, and complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be holomorphic on complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png.

Then

complex_analysis_7fa0559d56dcb4f1f930436c0aea2f89b7c6c393.png

Let:

Fix complex_analysis_4714803f526d076b85c9239f2b92578c110263b4.png, then

complex_analysis_26e3662ef294e804bd3fc6d5814fd5e2a469566a.png

where complex_analysis_afb1bc7ef6c183a97b917579e762d5b06c9009bf.png is the circle of radius complex_analysis_b4bc85fbe6b1085924f1ba268e01893b52ddad89.png centred at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png (Cauchy Integral formula).

complex_analysis_afb1bc7ef6c183a97b917579e762d5b06c9009bf.png is parametrized by complex_analysis_2d6b06dd3f397fc6025bcf5b65c2d1db3933a4bf.png, given by

complex_analysis_1e66a5221d20e79c417ca412d001528a0136dbdf.png

So, using the definition of the contour integral

complex_analysis_a69f93176010f961e1de21df66a5609522e7a09c.png

Suppose

complex_analysis_807ae749b511b065256ad39bfda91b7e8486e93f.png

i.e. maximum value is attained at the center of the circle. Then,

complex_analysis_8f2b829eac771dc4fb22a18f74c6fa985d1489cd.png

hence

complex_analysis_66bc4a931a8636afd274f0fee43779ddef86ae94.png

which, since complex_analysis_ea59e8fbaff6f9820f2d59bcbf3335e3b87ce0fe.png is clearly constant, we can use the linearity of the integral to get

complex_analysis_a6de4fff1df8df0037b1868c010fc4d8f7f51821.png

which is a non-negative continuous function, which implies that the expression inside the integral is zero for all complex_analysis_e11aba28b6489f1150660e9babb511e391f48239.png, i.e.

complex_analysis_ecf2d8f99471840ca35beac7eeccbe93fde583f9.png

which is true for every point in complex_analysis_0d4d108f7d692168da9e0c2407298eb51e388055.png is of this form, for some complex_analysis_858c7080eff8726480a7c6abb8344ee2e397af52.png, and some complex_analysis_e1c3809e67875f33e1f63f892a072bad78deff8e.png.

Hence complex_analysis_a91e8fdc353f84638328d9e24b685857a7dd77ee.png is constant on complex_analysis_0d4d108f7d692168da9e0c2407298eb51e388055.png.

Let complex_analysis_dc7e39d8e1b97836f49daa4a56d4b47d7855cea7.png, and complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be holomorphic and bounded on complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png, i.e.

complex_analysis_3d4eaee4ff8747f8c3b7d6850418e58e0ba846fd.png

for some complex_analysis_06c4a09d92b851bc0cf8a519f85459438f5959c4.png.

If complex_analysis_6c67b521afed5e7d8600f4f1b76d7b30a9cc8079.png achives its maximum at complex_analysis_c4f806c2ed58a3035e03bf180e0ca340e99a3aa9.png, then complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is constant on complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png.

Series Expansions for Holomorphic Functions

Stuff

Let complex_analysis_6f328629a606a1036804e6042351dac9e6bcf36a.png be a power series.

Then complex_analysis_77c6e14f777414682ca8f1ec79064b8c859abfd0.png s.t.

  1. complex_analysis_6f328629a606a1036804e6042351dac9e6bcf36a.png converges if complex_analysis_cca59b3383d9deb50c019999668b4bb39a429941.png
  2. complex_analysis_6f328629a606a1036804e6042351dac9e6bcf36a.png converges uniformly on complex_analysis_d44c385c95d64eac9cca6fd469277f66900651bc.png for all complex_analysis_74fbe06415533efe7bb560fbfb632ec67681ee3f.png
  3. complex_analysis_6f328629a606a1036804e6042351dac9e6bcf36a.png diverges if complex_analysis_8ae7e41baaba4859c77248e44697e5b5daaa8322.png

complex_analysis_6f328629a606a1036804e6042351dac9e6bcf36a.png is holomorphic on its disc of convergence complex_analysis_5d76d960685ac887d8a2c296a2641df3037fbef5.png, where complex_analysis_6dd4ec21f92a21e77e43374b4c7fd1cc8c2234c6.png is the radius of convergence.

To proof holomorphicity of convergent power series, one could do as follows:

Fix complex_analysis_e4fd228ea5cdc9ae354d5541abdcbc65b868b363.png. Then complex_analysis_440dac4814433073d75ee32003e344ee2fcf0004.png for some complex_analysis_858c7080eff8726480a7c6abb8344ee2e397af52.png. Series converges uniformly on complex_analysis_d44c385c95d64eac9cca6fd469277f66900651bc.png, thus partial sums are holomorphic, which implies series is holomorphic at complex_analysis_9c15196dd07b1add486b8b54592e74bfe946ed95.png.

Let complex_analysis_b4afb7b2eee809aae29f62450a15fecd0c356648.png be open, and complex_analysis_443078545102001f9f9f6cddfd4b98f32601f71b.png.

Then we say that complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is analytic if at everypoint complex_analysis_474617bdb7bb3e50af0bd51b0e1fd99f1ce8745a.png, complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png can be expressed as a convergent power series.

Suppose complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is holomorphic on complex_analysis_5d76d960685ac887d8a2c296a2641df3037fbef5.png. Then the Taylor series

complex_analysis_05e4ffb3a456b14d57283b7015f05b103665d921.png

converges to complex_analysis_3cde5172d50d013ad7e2a092d995a909e9739757.png on complex_analysis_5d76d960685ac887d8a2c296a2641df3037fbef5.png, and converges uniformly on complex_analysis_0d4d108f7d692168da9e0c2407298eb51e388055.png for all complex_analysis_a5e34cafa9f4147aba847de104ffb4a653800a03.png

where

complex_analysis_db79d4109c5b71a136b9a26c6a094ef5beee3c17.png

for any loop complex_analysis_6cd6d3a94855148ab982a9e17e08070f7bd81310.png with complex_analysis_338850f7e1cf3f1702f8138d97d1ca837e1af313.png.

It's REALLY important to realize that this Taylor series is only on some disk complex_analysis_5d76d960685ac887d8a2c296a2641df3037fbef5.png!!!

Therefore, if we know complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is holomorphic on some open subset complex_analysis_b4afb7b2eee809aae29f62450a15fecd0c356648.png, then if we want to talk about the Taylor series on any point complex_analysis_3e61f3c1ae92256436410aa5f5e37d4aa1228e48.png, still can only say something about this on some disk complex_analysis_bf0887e509278aa46cf5f4a2bdb5af263c30ed40.png!!

The Taylor series at some point complex_analysis_3158dce2e69e4bb12de085bd615b56c04a14cbdd.png is NOT necessarily the same as the Taylor series at some other point complex_analysis_0145ecdd65cb3abe2e2c66288b6aad04dc335cdf.png.

Let complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png be the circle of radius complex_analysis_9ab766fe2c081b82a304866d50f951fadbdb5704.png centered at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png where complex_analysis_4da0b9f60473be0304a5ba45afaa25f9ca3f08da.png. By CIF we know that for complex_analysis_f47cd937139d9e546569c920b4ac7f637fba2444.png we have

complex_analysis_5630f5b1b6b15977e61afa7394f80ea79cd80141.png

So

complex_analysis_4b4679340a2c61baa76bad0c254cf4a30b16d4f1.png

where we've used the fact that

complex_analysis_adc169d7e9d71ce114ffbb522cd7b74e88dc4aea.png

thus

complex_analysis_9b4388f550c357a83c03cce2d626894e2a31f13d.png

hence we could recognize the fraction above as the convergent series. Further, due to the series being convergent, we can interchange the summation and integration

complex_analysis_e41fc57c490e784a26b9e31b8e2816138a80e66f.png

which is just the Taylor series!

Taylor series of a complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png (if it exists) is unique, i.e. if complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is holomorphic on complex_analysis_5d76d960685ac887d8a2c296a2641df3037fbef5.png and

complex_analysis_f5ce33cd66458b57c99117ad4949e97f6d3293e0.png

A anulus is

complex_analysis_dfb30dba67a1df08fa6ff7008b38d4ec4fed327c.png

complex_analysis_6fb06e30babbe3a7a6bc300835bca72a30db06fa.png

Suppose complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is holomorphic on the anulus complex_analysis_4441052a5c292d26894bc2d9985bd3bf9c518318.png.

Then

complex_analysis_7b03c70cc4f73cf2c1b10383cfda05b556d55b0b.png

for any loop complex_analysis_f2df3b038eac6b761ed4a3fd14ba8081185b5c47.png with complex_analysis_34820caaef3e88539d086371967e9cc4522a5d66.png for all complex_analysis_c5de308ff5df60ab1f25776e1311334976587791.png, and the series converges uniformly on complex_analysis_d865fac4fdeb68b407a60bef3556dabf0bd09e6e.png for all complex_analysis_1ed50a8677f592c0e820d75714075c246c9c8ea9.png.

We say complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png is a singularity if complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is not holomorphic at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png.

Suppose complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png has a singularity at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png.

If there exists complex_analysis_8ae53a3d04627802bc79bbde92987929abdb717e.png s.t. complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is holomorphic on the punctured disk complex_analysis_4f9f3df94b9fe419aa1a7f29213ffe3022d31a94.png, complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png has a Laurent series centred at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png, valid on this disk

complex_analysis_e7da3c070d315684a37e1381c6aca9a4ce2c87cf.png

Then ONE of the following is true:

  1. complex_analysis_6b2b07726d97d06b8dfdb0654781ae9accdfeb56.png for all complex_analysis_895e94ab7124f6a395af67fe094ca759d6694c0b.png (complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png is a removable singularity)
  2. complex_analysis_7aa36cac93795e41c0b33aef54e9a9034a6bbedf.png such that complex_analysis_bf6c099c788fcc9c55f0695d29c2085df43cd13f.png but complex_analysis_daddda6b82567244ebd5bf8bd83883170cf29378.png for all complex_analysis_f1858ec10595278f3fb1ed96dbdf6649689fb556.png (complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png is a pole of order complex_analysis_fe52c58a96549528260fe0d7d04caf390dd47865.png)
  3. complex_analysis_ff1ba11a4391a09eb16435e2d3347fd2a1655aa0.png infinitively many negative complex_analysis_92483d0bb3bbad1c0b4630e7e529b0b5f3f8c48b.png s.t. complex_analysis_530c41a4ce95696e3c4da57bc668bfab644eda30.png (complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png is an essential singularity)

If you have a removable singularity and then consider the integral along some loop around complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png.

Then the integral is the integral along some loop on which complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is holomorphic, hence it's zero.

We can therefore redefine complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png to take on the value complex_analysis_a352ff1b37a5e98173f3ef2e177bdcf2965e10f0.png at the singularity point complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png, and we got ourselves a holomorphic function on the non-punctured disk centered at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png!

complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png is a zero of holomorphic function complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png if complex_analysis_ea59e8fbaff6f9820f2d59bcbf3335e3b87ce0fe.png.

Further, we say it's a zero of order complex_analysis_fe52c58a96549528260fe0d7d04caf390dd47865.png if

complex_analysis_ec5f808ceb9efbbbdfca310a20f515e48df2ffd3.png

but

complex_analysis_ffcf5126da6cb8b7273cf2da4bb51b456f39a07b.png

Then

complex_analysis_ac226541c484d691f8836d451c2222e2ab71cfe6.png

Let complex_analysis_5fbda07c45a608d0477cc50191ca91954f0ba2ca.png, complex_analysis_b4afb7b2eee809aae29f62450a15fecd0c356648.png be a neighbourhood of complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png, complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be holomorphic on complex_analysis_5760e806c8eb938d6a2563d076feec6ab9f9cafc.png, and such that complex_analysis_bb405ae1d0889ef17067696aa448d86b2f5fb753.png for a sequence of distinct points complex_analysis_6ac7ffcb46c55656f36bc62d1b8710f00ec78bb2.png which converge to complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png.

Then complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is identically zero on some disc centered at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png.

Let complex_analysis_fa5e4fac6adbec6d46728d93fd5fae757333cd67.png be holomorphic at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png be a zero of complex_analysis_bed561b338628a088ae69301d4bba9fd83a70bd2.png of order complex_analysis_fe52c58a96549528260fe0d7d04caf390dd47865.png.

Then complex_analysis_772c2fe4dfc89b11f234118f4a63f23cb32be1fd.png has

  1. has a pole of order complex_analysis_fe52c58a96549528260fe0d7d04caf390dd47865.png at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png if complex_analysis_8e54099c89f443ed0d65d73cec57fe7bd94070e3.png
  2. has a pole of order complex_analysis_cdb630a172354dd2e6bc9682b0a974c7055edabd.png (if complex_analysis_30a9470336817281cbe77252eb4e4df66670a4f6.png) at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png, if complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png is a zero of order complex_analysis_094b02afce734f4ce51933d0093ef3d2da9f8123.png of complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png, OR complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png is a removable singularity if complex_analysis_223ea5f159388a56ba9a417856db8804b9c4a0ae.png

Analytic continuation

Suppose complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is holomorphic on a domain complex_analysis_a14edc7d658dad8a2db23542aca549e15e4fc0d6.png with complex_analysis_505c692a1c47f7ff61cf4c8f738d78b09484e8e2.png and complex_analysis_7b2025be73038ef2a15eaa043ccee5634ff3dd73.png.

Then

complex_analysis_45dd0ad15b76bfdf560afcb2b74e41fb0bb877c1.png

Power series above might have radius of convergence complex_analysis_f1c0eb30f73aa6edc9e4cae2e7efdb7d29782178.png

complex_analysis_442ceee186486d5aae1f852fc55ebb1eef1c72b1.png

is a holomorphic on complex_analysis_5d76d960685ac887d8a2c296a2641df3037fbef5.png with complex_analysis_0a1ee5d8f5c633754e400c941e88bc2976cd3b99.png, i.e.

complex_analysis_7b26e39fd35cdbd37d67f02620389e7e57182417.png

One might then as does complex_analysis_74de6524033f3df0db27864260ee6bb11a075322.png on complex_analysis_72f801a5466f697121810037566bb9efd28156ae.png?

Yes, complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is an analytic continuation of complex_analysis_866ead89ec63f64bf1f7216d2aac4c2c37c2c921.png to complex_analysis_be71d447d3141967badf48942fb921fa12160b4a.png where

complex_analysis_259ca91c523aafe7457a0d11ef857626ca3c3544.png

and complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is well-defined.

Let complex_analysis_b7ac3b0898d8768b05377aba626154d5a492aceb.png be domain, and complex_analysis_4ef3444614b45065eb24938a20e418d3d6213c78.png be holomorphic.

We say that a holomorphic function complex_analysis_98e4350888d90e93b45394a201ddb3a348e20e35.png is an analytic continuation of complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png if

complex_analysis_6d8663c1b885833daafa1904863b65e7e7ca9be9.png

Let complex_analysis_dc7e39d8e1b97836f49daa4a56d4b47d7855cea7.png be a domain, complex_analysis_c4f806c2ed58a3035e03bf180e0ca340e99a3aa9.png, complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be holomorphic on complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png and such that

complex_analysis_b026c9bcfe4f1b020dc72faaf3c15a90cf389c07.png

for some complex_analysis_8ae53a3d04627802bc79bbde92987929abdb717e.png. Then

complex_analysis_dbe13618b1e62c24d3c9c1810968c1d53f825d16.png

Let complex_analysis_dc7e39d8e1b97836f49daa4a56d4b47d7855cea7.png be a domain, complex_analysis_c4f806c2ed58a3035e03bf180e0ca340e99a3aa9.png, complex_analysis_f86e72ce1b3a9443724af33cb5463b0dfda890ab.png be holomorphic and such that

complex_analysis_d5f24c748db6bedaad72db6d612013878fd4ad15.png

for some complex_analysis_8ae53a3d04627802bc79bbde92987929abdb717e.png.

Then complex_analysis_90d3330f944354bbf9ca93c70d3cc84e47b4e47d.png for all complex_analysis_e87fb4ed76655c50da322f977d89a150edb9d855.png.

Let complex_analysis_dc7e39d8e1b97836f49daa4a56d4b47d7855cea7.png be a domain, complex_analysis_5fbda07c45a608d0477cc50191ca91954f0ba2ca.png, and complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be holomorpic on complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png and such that

complex_analysis_cae8d4e7c8bc45c2da06c23f62107c3e9bbe7241.png

for a sequence of distinct points complex_analysis_8520901d906bc8436719f469bb9697ff17babb2a.png which converge to complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png. Then

complex_analysis_dbe13618b1e62c24d3c9c1810968c1d53f825d16.png

Let complex_analysis_dc7e39d8e1b97836f49daa4a56d4b47d7855cea7.png be a domain, complex_analysis_5fbda07c45a608d0477cc50191ca91954f0ba2ca.png, and complex_analysis_fa5e4fac6adbec6d46728d93fd5fae757333cd67.png be holomorphic on complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png and such that

complex_analysis_e90e65f88938dfd3f7a2f827d7fca60787b6a6aa.png

for a sequence of distinct points complex_analysis_8520901d906bc8436719f469bb9697ff17babb2a.png which converge to complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png. Then

complex_analysis_09073a17addf68807b50291ee996423fad719819.png

Let complex_analysis_fa5e4fac6adbec6d46728d93fd5fae757333cd67.png be holomorphic at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png, where complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png is zero of order complex_analysis_fe52c58a96549528260fe0d7d04caf390dd47865.png. Then

  1. if complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png is not zero of complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png, then complex_analysis_772c2fe4dfc89b11f234118f4a63f23cb32be1fd.png has a pole of order complex_analysis_fe52c58a96549528260fe0d7d04caf390dd47865.png at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png
  2. if complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png is a zero of order complex_analysis_094b02afce734f4ce51933d0093ef3d2da9f8123.png of complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png, then complex_analysis_772c2fe4dfc89b11f234118f4a63f23cb32be1fd.png has a pole of order complex_analysis_cdb630a172354dd2e6bc9682b0a974c7055edabd.png at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png if complex_analysis_e005f3418486c2c3d5f14f1275d5c39abc5981fc.png, and has a removable singularity at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png otherwise.

Cauchy Residue

Theorem

Let complex_analysis_5fbda07c45a608d0477cc50191ca91954f0ba2ca.png, complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be holomorphic on the punctured disc complex_analysis_1adad2960e179b78aff9f14fe69148b59014cbbf.png for some complex_analysis_8ae53a3d04627802bc79bbde92987929abdb717e.png, with an isolated singularity at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png, and complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png be a loop inside complex_analysis_1adad2960e179b78aff9f14fe69148b59014cbbf.png, with complex_analysis_338850f7e1cf3f1702f8138d97d1ca837e1af313.png. Then

complex_analysis_ec29eff10b5c67d30e4c315a644f527633a8da24.png

where complex_analysis_6f0abbdecd925d85c39899f69039712051821826.png is the coefficient of the complex_analysis_2afc8d924db8e7aa2e9d53446d704cd6be4f608f.png term in the Laurent expansion of complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png centred at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png,

complex_analysis_8113030a72e1abf5ada12678d5fea50f4f20dd77.png

Let complex_analysis_5fbda07c45a608d0477cc50191ca91954f0ba2ca.png and complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be holomorphic on the punctured disc complex_analysis_1adad2960e179b78aff9f14fe69148b59014cbbf.png, for some complex_analysis_8ae53a3d04627802bc79bbde92987929abdb717e.png, with an isolated singularity at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png.

Then the residue of complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png, is

complex_analysis_d0cc5ec2aced113eaa230ac76b5cb2b5ca5b29fd.png

where complex_analysis_b4fa82353c422fbd953e8d17bc7d8f76506d52c5.png is the term in the Laurent series of complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png centered at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png.

Let complex_analysis_5fbda07c45a608d0477cc50191ca91954f0ba2ca.png, and complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be holomorphic on the punctured dist complex_analysis_1adad2960e179b78aff9f14fe69148b59014cbbf.png for some complex_analysis_8ae53a3d04627802bc79bbde92987929abdb717e.png, with removable singularity at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png. Then

complex_analysis_6a98a075dafef4cde7b41cf5768ff03791abee70.png

Let complex_analysis_5fbda07c45a608d0477cc50191ca91954f0ba2ca.png and complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be holomorphic on the punctured disc complex_analysis_1adad2960e179b78aff9f14fe69148b59014cbbf.png, for some complex_analysis_8ae53a3d04627802bc79bbde92987929abdb717e.png, with a pole of order complex_analysis_fe52c58a96549528260fe0d7d04caf390dd47865.png at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png. Then

complex_analysis_cfdfbc17d6098ed2bfb683c1eae7b8a5149be9c8.png

Let complex_analysis_5fbda07c45a608d0477cc50191ca91954f0ba2ca.png and complex_analysis_bed561b338628a088ae69301d4bba9fd83a70bd2.png, complex_analysis_3807b8495f01004f1d88370e2f4fad2f5032db06.png be holomorphic on complex_analysis_1adad2960e179b78aff9f14fe69148b59014cbbf.png for some complex_analysis_8ae53a3d04627802bc79bbde92987929abdb717e.png, such that complex_analysis_3807b8495f01004f1d88370e2f4fad2f5032db06.png has a simple zero (complex_analysis_4d056f7330d032cc6971fcc8578a75f025a39725.png) at complex_analysis_17b01f3397a8d65043a65bbb00dbe463e6de6331.png, while complex_analysis_c3f49dcce5439213174eb266cd3a36b32d362787.png. Then, defining complex_analysis_f1ebcdd8cee44d9aaeade384b942b2315d9158a5.png we have

complex_analysis_bf6a67307e803f14cbbc64bb974964eee6fee593.png

Let complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png be a loop, and complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be holomorphic inside and on complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png except for finitely many isolated singularities complex_analysis_cb8c195d3ef5c37848351125171df7877720f08f.png. Then

complex_analysis_e0963f500cd503cdab566a6458a6376609c8124f.png

Let complex_analysis_dc7e39d8e1b97836f49daa4a56d4b47d7855cea7.png be a domain.

A function complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is meromorphic on complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png if for all complex_analysis_e87fb4ed76655c50da322f977d89a150edb9d855.png, either complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png has a pole at complex_analysis_9c15196dd07b1add486b8b54592e74bfe946ed95.png or complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png is holomorphic at complex_analysis_9c15196dd07b1add486b8b54592e74bfe946ed95.png.

Application: trigonometric integrals

Integrals of the form

complex_analysis_fd4d40e31b69b8b033c15ae7fb0de3284febaeec.png

for ration function complex_analysis_6dd4ec21f92a21e77e43374b4c7fd1cc8c2234c6.png, can often be evaluated by considering a contour integral of appropriate function around the unit circle centered at complex_analysis_d83ba0b34682ac87f8d84c8310f6bbd28d1fe65b.png.

On complex_analysis_323749d66bbf3ba072156aea554e0d4a5b2294a1.png with complex_analysis_f7cc79c63fc6060c8eb0d6926134307618f868e8.png we have

complex_analysis_7a4d2394447b71d085d57dc2116e0431366cf4b9.png

and

complex_analysis_ef638b1238610674841cfdbdf0baaa3841d35a15.png

Defining complex_analysis_c22b9bda3f71f4ec7426e8046743b5e2c86ef093.png, we therefore have that

complex_analysis_7d3ed4b610bf1de509c4766f648c05939498ecf8.png

If complex_analysis_07118d61e7e8d5b7367f147b0ee133867b48c5e4.png parametrized by complex_analysis_94de5bb8f0d2a9ef4b9e87635319ba4f7e12c984.png defined by complex_analysis_a6af9ccbbe63390bab46bbd4054f52f20e6dc9e4.png, then

complex_analysis_8f3857a8035ed3adb409bae21278e91cade2dbca.png

It basically comes down to rewriting complex_analysis_a1f577a43ce32a53d2309c84b4c0639db6e501d8.png and complex_analysis_6caffdc854576e93c413345b24d375689db31ee3.png with complex_analysis_f7cc79c63fc6060c8eb0d6926134307618f868e8.png, which then often provides us with a rational function of which it's substantially esaier to obtain the singularities, thus the residue, hence the integral around complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png.

Improper Integrals

We define the Cauchy principal value of the integral

complex_analysis_2a2b5bf53e4e939c2608835a1acbef170b826c74.png

as

complex_analysis_9bb678cc0b6ddb086509ba31f9c7722d9eb7bf54.png

Let complex_analysis_bfee8369178158db8d7174a3ca5e20f3a04662d3.png be a rational function, where

complex_analysis_cffde95a4587ad2ba1f01c26f73494fb437bb982.png

ad complex_analysis_8276fba3b5a671a38a120ae441dcd9cd1f830053.png such that complex_analysis_5a3cb9b7f0622abd7a12c38177154de744348096.png. Then

complex_analysis_8cf31b222cd34174406ab576891682c63f64ec72.png

where complex_analysis_32f0951f46a5d292b3202644b07ff1c6b39d753f.png and complex_analysis_f86fa31b0e716a6e57d10bbd67ad345a2621cbbb.png are the semicircular contours from complex_analysis_7f1cd3d90b6abe0c32a765d3d8b3950df906296d.png to complex_analysis_9ab766fe2c081b82a304866d50f951fadbdb5704.png in the upper and lower half-plane, respectively.

Let

  • complex_analysis_dc7e39d8e1b97836f49daa4a56d4b47d7855cea7.png be a domain
  • complex_analysis_c4f806c2ed58a3035e03bf180e0ca340e99a3aa9.png, complex_analysis_cdd1cc131da6040eca078917132a377727053c44.png be meromorphic on complex_analysis_b689cba8d7566f6adaf605a844e193a27e155078.png with a simple pole at complex_analysis_39f378a2afcec6357cdf96759a145bf72beeb14b.png
  • complex_analysis_deb316b87c62843597daddd9ebbbd10e7eae1c9e.png be the circular arc parametrized by complex_analysis_37e62daa197a3bbde694adb591619c788f6b68cf.png for complex_analysis_a73df149805be81a5c9a250dd0ebda6965509dd7.png for some complex_analysis_d6fa32febd839644d26168a1534a0b1be0625e61.png.

Then

complex_analysis_7e6e390d41453c06d7c168f810d6e2409705eb3c.png

Let complex_analysis_e54e0f2d4138cbdba917bf64636de249daba706e.png, and let complex_analysis_3782423da1928f706d96b41543893179a3a636c3.png be a loop with complex_analysis_d40494e5bd6154a893679b7631692c8b80786121.png. Then

complex_analysis_edce51c7e133f0ed4bf480e2c5a361da68ac36d6.png