THRY OF FUNC OF CMPLX VARIABLE
THRY OF FUNC OF CMPLX VARIABLE M 361
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Date Created: 09/06/15
Outline Based on Complex Variables and Applications Brown and Churchill7 6th ed Alex Winbow December 19 2001 1 Chapter 1 7 Basics 2 Chapter Two Analytic Functions De nition of limit limzAZO we implies that 7 wol lt 6 Whenever 0 lt 2 7 20 lt 6 CauchyRiemann Equations 1f fz uxy ivxy and exists 20 10 iyo then the first order partial derivatives of u and f must exist at zoy0 and they must satisfy the CauchyRiemann equations ugc vy uy 7ng Sufficient Conditions for Differentiability Let the function yzy ivzy be de ned throughout some e neighborhood of 20 104in Suppose that the first order partial deriva tives of the functions u and V With respect to x and y exist everywhere in that neighborhood and that they are continuous at 10 yo Then ifthose partials derivatives satisfy the CauchyRiemann equations at 10 yo the derivative existsi Polar Coordinates Same as theorem in 18 polarizedi Polar form of CR 1 ur 5 ug ivr 29 Analytic Functions A function f of z is analytic in an open set if it has a derivative at each point in that set In particular fin analytic at a point 20 if it is analytic in a neighborhood of zoi 0 An entire function is a function that is analytic at each point in the entire nite plane 0 If a point fails to be analytic a point 20 but is analytic at some point in every neighborhood of 20 then 20 is called a singular point 0 Since the derivatives of the sum and product of two functions exists wher ever the functions themselves have derivatives the sum and product of two analytic functions are themselves analytic So is the quotient wherever the denominator does not vanish So is a composition 0 Reflection Principle suppose f is analytic in some domain D which con tains a segment of the Xaxis and is symmetric to that axis lff fx is real for each point X on the segment then for each point z in the domain f2 210 Harmonic Functions 0 realvalued function H of two real variables X and y is said to be harmonic in a given domain of the Xy plane if throughout that domain it has con tinuous partial derivatives of rst and second order and satis es the PDE known as Laplace s equation Hm Hyy 0 o If a function uzy ivzy is analytic in a domain D then its component functions u and v are harmonic in D o If two given functions u and v are harmonic in D and the FOPD satisfy CR throughout D v is said to be a harmonic conjugate of u o A function uzy ivzy is analytic in a domain D iff v is a harmonic conjugate of u 3 Chapter 3 7 Elementary Functions 0 sinz so m 8787 2 o sinhz o coshz my 323 The Logarithmic Function and Its Branches o logz lnlzl iang Logz ln 23914ng 0 logz Logz 2in7r o branch of a multiplevalued function f is any singlevalued function F that is analytic in some domain at each point z of Which the value Fz is one of the values fz o The function Logz 1nquot 23914ng 7 gt 07 lt Ang lt 7r is called the principal branch 0 A branch cut is a portion of a line or curve that is introduced in order to de ne a branch F is a multiplevalued function f Points on the branch cut for F are singular points7 and any point that is common to all branch cuts of f is called a branch point 324 Complex Exponents When 2 y 0 and the exponent c is any complex number7 20 is de ned as 20 ec ogz Principal value de ned as obvious 4 Chapter Four Integrals 431 Complexvalued Functions Wt o Meanvalue theorem for derivatives doesn7t apply 0 Integrals generally exist if piecewise continuous 432 Contours 0 An arc is a simple or Jordan arc if it does not cross itself 0 Similarly7 simple closed curve or Jordan curve If the derivatives of the component functions of an arc exist and are con tinuous7 the arc is differentiable o smooth are has a continuous derivative on the closed interval and is nonzero on the open interval 0 contour is a piecewise smooth arc o A simple closed contour has only the beginning and end points the same 433 Contour Integrals 0 De nition of line or contour integral 1 fztztdt S ML C a 434 Antiderivatives 0 An antiderivative is necessarily an analytic function Theorem Suppose that a function fis continuous on a domain D If any one of the following statements true then so are the others 7 F has an antiderivative F in D 7 The integrals of along contour ling entirely in D and extending from any xed point 21 to any xed point 22 all have the same value 7 The integrals of around closed contours lying entirely in D all have value zero Basically all are true or none are true CauchyGourst Theorem If a function f is analytic at all points interior to and on a simple closed contour C then f0 0 435 Proof of the Theorem 7 Simply and MultiplyConnected Domains 7 A simply connected domain D is a domain such that every simple closed contour within it encloses only points of D 7 A domain that is not simply connected is multiply connected 7 Theorem If a function fis analytic throughout a simply connected domain D then f0 0 7 Corollary lf C1 and C2 denote POSCC where C2 is interior to C1 and if function f is analytic in the closed region between and including the two contours then the integral of f around C1 equals the integral of f around C2 Principle of Deformation of Paths 436 Cauchy Integral Formula 0 Let f be analytic everywhere within and on a simple closed contour C taken in the positive sense If 20 is any point interior to C t en f20 437 Derivatives of Analytic Functions Theorem if a function is analytic at a point then its derivatives of all orders are also analytic functions at that point Corollary If a function is analytic at a point then the component func tions uv have continuous partials of all orders at that point 438 Liouville7s Theorem and the Fundamental The orem of Algebra If f is entire and bounded in the complex plane then fz is constant throughout the plane Fundamental Theorem of Algebra Cauchy7s inequality f zol S 7Lng fzo i 02 f 20 eww 439 Maximum Moduli of Functions Theorem if a function f is analytic and not constant in a given domain D then has no maximum value in Corollary essentially maxima of must occur on the boundary 5 Chapter Five 7 Series Convergence of Sequences and Series Convergence absolute convergence Necessary but not suf cient that 2 A 0 Taylor Series Suppose that a function fis analytic throughout an open disk zizol lt R0 Then at each point z in that disk fz has the series representation that is the power series converges to fz Whenever 2 7 20 lt R0 Where an 5 43 Laurent Series Suppose that a function f is analytic throughout an annular domain R1 lt l2 7 20 lt R2 and let C denote any positively oriented simple closed contour around 20 and lying in that domain Then7 at each point z in the domain7 has the series representation 220 an 2 7 20 7 1 f 2 dz Where an 7 c w L 2 dz 27m f0 2720 1 f bn C71 2 dz 27m 0 2720 n 544 Examples Useful Series 00 12 Zenm 71 n0 545 Absolute and Uniform Convergence of Power Se ries If a power series centered at 20 converges at some point 21 then it is absolutely convergence at each point 2 in an open disk centered at 20 extending out to 21 in radius Corollary to an unmentioned theorem a power series represents a con tinuous function Sz at each point inside its circle of convergence 546 Integration and Differentiation of Power Series Power series can be integrated termbyterm7 and summations pulled in and out of integrals Corollary the sum of a power series is analytic at each point z interior to the circle of convergence of the series A power series can be differentiated term by term 547 Uniqueness of Series Representations o If a power series converges to fz at all points interior to some Circle l2 7 20 R then it is the Taylor series for f 0 Likewise for Laurent series7 but necessarily only in an annular domain about 20 o Multiplication and Division of Power Series Leibniz7s rule 220 Zgtfkgnkz n 7 nl k 7 kln7 k On Zakbnik k0 1 n On b an Zbkcnik 0 k1 6 Chapter Six Residues and Poles 7 Residues o A singular point is isolated if in addition there is a deleted neighborhood throughout which fis analytic 1 2720 o The residue of f at z is the coefficient of in an Lseries of f at z 71 Residue Theorems 0 Cauchyls Residue Theorem Let C be a POSCC If a function fis analytic inside and on C except for a nite number of singular points then the value of the integral around C is 2i7r the sum of the residues 0 Sometimes if the function is analytic at each point in the nite plane exterior to C it is more ef cient to evaluate the intergral of f around C by nding a single residue 0 Theorem If a function fis analystic everywhere in the nite plane except for a nite number of singular points interior to a POSSC C then l l fltzgtdz 27r2Reso 2fl c 2 2 72 Three types of singular points 0 Three types of isolated singular points 0 Pole of order m where m is the highest power of 2 7 20 in the Laurent series o Removeable singular point when all the bls are zero so the residue is zero 0 Essential singular point when an in nite number of b7s are nonzero All hell breaketh loose taking on every nite value possibly except zero 73 Residues at Poles 0 An isolated singular point 20 of a function f is a pole of order m iff fz can be written 3 where is nonzero mil Moreover Resz20fz o ls is always true that if 20 is a pole of a function f then limznzO 00 While the theorem can be useful often it is better to write directly as Laurent series 74 Zeros and Poles of Order In A function that is analytic at a point 20 has a zero of rder In there iff there exists a function g Which is analytic and nonzero at 20 such that 162 Z Zom9z Zeros of order m are sources of poles of order In Theorem as a result Corollary If p and q analytic at point 20 iff pzo 07 q20 07 and til20 y 0 Res zomz we 42 Higher order formulae exit but are not practical 4 20 75 Conditions under which fz E 0 If 0 at each point z of a domain or arc containing a point 20 then E 0 in any neighborhood N0 of 20 throughout Which f is analytic That is7 0 at each point z in N0 Theorem ifa function if analytic throughout a domain D7 and the function is zero at each point of a subdomain or arc inside D7 then the function is zero throughout D Corollary a function that is analytic in a domain D is uniquely determined over D by its values over a domain or along an arc contained in D 76 Behavior of f Near Removeable and Essential Sin gular Points A function is anlways analytic and bounded ins ome deleted neighborhood of a removeable singularity Suppose a function is analytic and bounded in some deleted neighborhood of a point 20 if f is not analytic at 20 then its a removeable singularity Essential singularity not only does hell break loose7 but the function assumes values arbitrarily close to any given number 8 Chapter Seven 7 Applications of Residues Distinction between pair of improper and Cauchy RV integrals lf CPV converges not necessarily true that other does Method of evaluating improper integrals 861 Improper Integrals Involving Sines and Cosines Jordan7s inequality OW e RSinedt lt R gt 0 862 De nite integrals involving sines and cosines 863 Indented paths 864 Integration Along a Branch Cut 865 Argument PRinciple and Rouche7s THeorem Meromorphic E analytic in a domain except possibly poles Winding number theorem Z 7 P Roche s Theorem Let two functions f and g be analytic inside and on a SCC C7 and suppose that gt at each point on G Then and 92 have the same number of zeros7 counting multiplicities inside C 866 Inverse Laplace Transforms Forward transform Fs fem e ftdt Backward transform limRH00 fLR 5 Fsds t gt 0 Backward transform PV 5 Fsds t gt 0 1 251 R6552 65478 Res55O 5 FSRes5 e Fsl 26 Reei b1t z tm ll tgt0
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