Introduction to Complex Analysis
Introduction to Complex Analysis MATH 185
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Date Created: 10/22/15
MATH 185 THEOREMS Dave Penneys Summer 2009 Here is your arsenal of theorems with which to bombard problems For these theorems unless otherwise stated assume 1 D is a domain 2 f D a C is continuous and 3 7 ab a D is a contour Theorem 1 Extreme Value Suppose S C D is compact and g D a R is continuous Then there are 2021 6 S such that for every 2 E S 9 S 92 S 9217 ie g attains its maccimum and minimum on S Theorem 2 Cauchy Riemann Suppose f is di erentiable at 20 Then the partial deriva tives of uv eccist at 20 and um20 vy20 and uy20 7vw20 This can be expressed in polar coordinates if 20 31 0 as ru20 11920 and ue20 err7a Theorem 3 Suf cient Condition for Differentiability Suppose f u iv and there is a 20 E C and an 8 gt 0 such that I uv have continuous partial derivatives in Bg20 and 2 the Cauchy Riemann equations are satis ed at 20 rectangular or polar if 20 31 0 Then f is di erentiable at 20 Theorem 4 Harmonic Conjugates f u iv is holomorphic on D if and only ifv is a harmonic conjugate ofu on D Note We need to know that iff u iv is holomorphic on D then uv have continuous second order partial derivatives on D We have not proven this yet Theorem 5 Reparametrization dz is independent of reparametrization ie if 7 15 046 a ab is di erentiable such that t gt 0 for all t then fv tvo t dt fvtv t di Theorem 6 Integral Bounding Suppose M gt 0 such that S M for all z E im y and L is the length of y Then fz dz ML Theorem 7 Fundamental Theorem of Calculus If P V on ab and P is continuous on db then b yt dt m 7 Pa Corollary If F f on D then f2 dz Fvb e Fva and the integral is independent of path from 7a to yb Theorem 8 Antiderivative The following are equivalent 1 f has an antideriuatiue on D ie there is a holornorphic function F on D such that F f on D and 2 For every closed contour y C D dz 0 7 Theorem 9 Rectangle Lemma Suppose R is a rectangle contained in D and suppose P 3R ff is holornorphic on D then fz dz 0 Theorem 10 Simply Connected Domain Suppose D is simply connected and f is holo rnorphic on D Then f has an antideriuatiue on D Theorem 11 Cauchy Goursat Suppose f is holornorphic on D Then for all simple closed contours V such that ins y C D fz dz 0 7 Theorem 12 Deformation Suppose f is holornorphic on D and 7172 are two contours in D such that either 1 7172 have the same endpoints or 2 7172 are simple elosed If yl may be quotcontinuously deformed into 72 in D then fz dzfz dz Theorem 13 Cauchy7s Integral Formula Suppose f is holomorphie on D V is a positively oriented simple elosed contour and 20 E ins y C D Then 1 f z f20 dz 27ri z 7 20 7 Theorem 14 Cauchy7s Differentiation Formula Suppose f is holomorphie on D V is a positively oriented simple elosed contour and 20 E ins y C D Then f has continuous derivatives of all orders in D and fn20 2 z 7 20 1 Moreover iff u iv then uv have eontinuous partial derivatives of all orders in D Note This nishes the proof of the Harmonie Conjugates Theorem Theorem 4 Theorem 15 Morera 1 Suppose fz dz0 for all elosed y C D Then f is holomorphie on D Theorem 16 Morera 2 Suppose fz dz0 for all simple elosed y C D such that ins y C D Then f is holomorphie on D Theorem 17 Derivative Bounding Suppose f is holomorphie on D Suppose 20 E D and R gt 0 Then on CR20 the circle of radius R centered at 20 for all z E CR20 VM on lt71 R lf 2N Rn where MR rnaxlfzlz E CR20 Theorem 18 Liouville A bounded entire function is constant Theorem 19 Fundamental Theorem of Algebra Every compled polynomial of degree greater than or equal to one has a root Theorem 20 Maximum Modulus Principle Suppose f is holomorphic on D and there is a 20 E D such that S lf20l for all z E D Then f is constant on D Theorem 21 Local Uniform Convergence Suppose fn a f locally uniformly on D and fn is holomorphic on D for all n E N Then f is holomorphic on D Theorem 22 Convergence of Power Series Suppose 00 Z anz 7 20 n0 converges for all z E D BR20 for some R gt 0 Then I the sum converges absolutely on D and 2 the partial sums converges locally uniformly on D to a holomorphic function Corollary Analyticity implies holomorphicity Theorem 23 Taylor Suppose f is holomorphic on D BR20 for some R gt 0 Then there is a unique power series representation W 020 mezif lee20gt which converges on D Corollary Holomorphicity implies analyticity Theorem 24 Identity Suppose fg are holomorphic on a domain D and suppose is a sequence in D such that 2k a 20 E D If fzk gzk for all h E N then f g on D Theorem 25 Riemann7s Principle Suppose 20 is an isolated singularity of the holomorphic function f ff is bounded in a deleted neighborhood ofzo then 20 is a removable singularity Theorem 26 Casorati Weierstrass Suppose 20 is an isolated essential singularity of the holomorphic function f and U C D is a deleted neighborhood of 20 Then fU is dense in CC Theorem 27 Convergence of Laurent Series Suppose converges for all z E D z E CR1 lt l2 7 zoerg for some 0 2 R1 lt R2 3 00 Then 4 I the sum conuerges absolutely on D and 2 the partial sums conuerges locally uniformly on D to a holomorphic function Theorem 28 Laurent Suppose f is holomorphic on z E D z E CR1 lt z 7 z0 lt r2 for some 0 3 R1 lt R2 3 00 Then there is a unique Laurent series representation 00 W Z w 7 20 which converges on D where for each n E Z 1 f w an w 7 zo 1 7 dz where y C D is any positiuely oriented simple closed contour with zo E insD Theorem 29 Cauchy7s Residue Suppose f is holomorphic on D ecccept at nitely many points z1 zn and y C D is a positively oriented simple closed contour such that z E ins y for allj 1 n Then 1 71 dz Reszzjfz 7 Theorem 30 Jordan7s Lemma Suppose PR is the positively oriented semicircle of radius R and suppose f is holomorphic on the upper half plane outside PRO for some R0 gt 0 and continuous on the boundary Suppose further that for all z 6 PR we have 3 MR for some MR 2 0 and that MR 7 0 as R 7 00 Then for all a gt 0 fzei dz 7 0 as R 7 00 FR Theorem 31 Argument Principle Suppose f is meromorphic in D and y C D is a positively oriented simple closed contour which avoids the zeroes and poles off ff is not the zero function then I There are nitely many zeroes and poles off inside 7 and 1 2 dz Z 7 P where Z is the number of zeroes and P is the number of poles in z 7 off inside 7 including multiplicity Theorem 32 Rouch Suppose fg are holomorphic in D and y C D is a positively oriented simple closed curue If gt for all z E im y then f and f g have the same number of zeroes inside 7 including multiplicity Theorem 33 Open Mapping Suppose f is holomopphie and noneonstant on D andU C D is open Then fU is open Theorem 34 Schwartz7s Lemma Suppose f B10 a B10 is holomopphie and f0 0 Then 1 WZ S M and 2 f 0 1 with equality in eitheiquot if and only if fz ewz foiquot some 6 E 027T Theorem 35 Conformal Mapping The following are equivalent foiquot f on D I f is holomoiphie and f z 31 0 foiquot all z E D 2 f is holomopphie and locally injective ie foiquot every 2 E D there is an 8 gt 0 such that f is injective on Bgz and 3 f is conformal and Cl
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