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# Class Note for MATH 322 at UA

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This 8 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 12 views.

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Date Created: 02/06/15
Limits contir Chapter 13 Complex Numbers Sections 133 amp 134 Chapter 13 Complex Numbers Function of a complex variable Limits continuity and differentiation Limits and continuity A for 1y Differentiability Analytic functions ifl Funtion of a complex variable 0 A singlevalued function f of a complex variable 2 is such that for every 2 in the domain of definition D of f there is a unique complex number W such that W fz o The real and imaginary parts of f often denoted by u and V are such that fzuX7yiX7y7 2Xiy7 X7y R7 fz ltC7 with uXy E R and VXy E R Chapter 13 Complex Numbers Function of a complex variable Limits continu and differentiation Limits and continuity erion for analyticity Differentiability Analytic functions Function of a complex variable continued 0 Examples 0 f2 z is such that uxy x and Vxy y c Find the real and imaginary parts of f2 Z 1 o f2 E IS defined for all z 75 0 and IS such that X y vx 2 y X2y2 Chapter 13 Complex Numbers Functior rf 39 complex variable Limits continuity and differentiation Limits a ntinuity A for ici ty Differentiability Analytic functions 2 Limits and continuity 0 An open neighborhood of the point 20 E C is a set of points z E C such that z zol lt 67 for some 6 gt O 0 Let f be a function of a complex variable 2 defined in a neighborhood of z 20 except maybe at z 20 0 We say that f has the limit W0 as 2 goes to 20 ie that lim f2 2 W07 ZgtZO if for every 6 gt 0 one can find 6 gt 0 such that for all z E D z zo lt6gt fz wol lt6 2i z2 1 0 Example llm zgtl Z I Chapter 13 Complex Numbers Function of a complex variable Limits continuity and differentiation Limits and continuity A criterion for analyticity Differentiability Analytic functions Continuity o The function f is continuous at z 20 if f is defined in a neighborhood of 20 including at z 20 and lim f2 fzo Zgtzo o If fz is continuous at z 20 so is fz Therefore if f is continuous at z 20 so are 5Ref 3m and f2 0 Conversely if uXy and VXy are continuous at X07y0 then f2 uXy iVXy with z X iy is continuous at 20 2 X0 l iyo 0 Example Is the function such that f2 m22z2 for z 75 O and f0 0 continuous at z 0 Chapter 13 Complex Numbers Functir f a complex variable Limits continuity and differentia n Lil39nit ncl continuity 1 Differentiability Analytic functions 3 Differentiability 0 Assume that f is defined in a neighborhood of z 20 The derivative of the function f at z 20 is fzo Iim W 2 m 2 f20 AZHO AZ zgtzo 2 20 7 assuming that this limit exists o If f has a derivative at z 20 we say that f is differentiable at z 20 0 Examples 0 f2 2 is continuous but not differentiable at z 0 o f2 z3 is differentiable at any 2 E C and fz 322 Chapter 13 Complex Numbers Function of a complex variable Limits continu and differentiation Limits and continuity erion for analyticity Differentiability Analytic functions Rules for continuity limits and differentiation 0 To find the limit or derivative of a function fz proceed as you would do for a function of a real variable 0 Examples 1 1 f 2 22 o zquotnzquot 1 nGN c Find lim 2l 247139 2 Chapter 13 Complex Numbers rf a complex variable Limits continuity and differentiation continuity A for ici ty Differentiability Analytic functions Rules for continuity limits and differentiation continued 0 Properties involving the sum difference or product of functions of a complex variable are the same as for functions of a real variable In particular 0 The limit of a product sum is the product sum of the limits 0 The product and quotient rules for differentiation still apply 9 The chain rule still applies 0 Examples 2 Find 12 dz 2 d 4 0 Find Ez392 7 Chapter 13 Complex Numbers Function of a complex variable Limits continuity and differentiation Limits and continuity A criterion for analyticity Differentiability Analytic functions 4 Analyticuni 0 A function fz is analytic at z 20 if fz is differentiable in a neighborhood of 20 o A region of the complex plane is a set consisting of an open set possibly together with some or all of the points on its boundary 0 We say that f is analytic in a region R of the complex plane if it is analytic at every point in R 0 One may use the word holomorphic instead of the word analytic Chapter 13 Complex Numbers Functir f a complex variable Limits continuity and differentia n Limit ncl continuity 1 Differentiability Analytic functions Analytic functions continued 0 A function that is analytic at every point in the complex plane is called entire 0 Polynomials of a complex variable are entire 0 For instance f2 32 722 z3 is analytic at every 2 o Rational functions of a complex variable of the form z f2 where g and h are polynomials are analytic z everywhere except at the zeros of hz z2 1 o For Instance Z I IS analytic except at z I o In the above example 2 i is called a pole of fz Chapter 13 Complex Numbers Limits continuity and differentiation The CauchyRiemann equations A criterion for analyticity Harmonic functions 5 The Cauhy Riemann equations o If f2 uXy iVXy is defined in a neighborhood of z x iy and if f is differentiable at 2 then uXX7y VyX77 and uyX7ylXX7y These are called the CauchyRiemann equations 0 Conversely if the partial derivatives of u and V exist in a neighborhood of z X iy if they are continuous at z and satisfy the CauchyRiemann equations at 2 then f2 uXy iVXy is differentiable at z o The CauchyRiemann equations therefore give a criterion for analyticity Chapter 13 Complex Numbers Limits continuity and x The CauchyRiemann equations A criterion for Harmonic functions 0 Indeed if a function is analytic at 2 it must satisfy the CauchyRiemann equations in a neighborhood of 2 In particular if f does not satisfy the CauchyRiemann equations then f cannot be analytic 0 Conversely if the partial derivatives of u and V exist are continuous and satisfy the CauchyRiemann equations in a neighborhood of z X iy then f2 uXy iVXy is analytic at z 0 Examples 0 Use the Cauchy Riemann equations to show that 2 is not analytic 1 9 Use the Cauchy Riemann equations to show that IS analytic z everywhere except at z 0 Chapter 13 Complex Numbers Limits continuity and differentiation The CauchyRiemann equations A criterion for analyticity Harmonic functions Application of the Cauchy Riemann equations 0 A consequence of the CauchyRiemann equations is that fz uX ivX vy iuy 2 0 We will use these formulas later to calculate the derivative of some analytic functions 0 Another consequence of the CauchyRiemann equations is that an entire function with constant absolute value is constant In fact a more general result is that an entire function that is bounded including at infinity is constant Chapter 13 Complex Numbers Limits continuity and d o The CauchyRiemann equations A criterion for analyticity Harmonic functions 6 Harmonic functions 0 One can show that if f is analytic in a region R of the complex plane then it is infinitely differentiable at any point in R o If f2 uXy iVXy is analytic in R then both u and V satisfy Laplace s equation in R ie V2u2uXX l uyy207 and VZV VXX lVyyZO 3 o A function that satisfies Laplace s equation is called an harmonic function Chapter 13 Complex Numbers Limits continuity and differentiation The Cauchy Riemann equations A criterion for analyticity Harmonic functions Harmonic conjugate o If f2 uX7y iVX7j is analytic in R then we saw that both u and V are harmonic ie satisfy Laplace s equation in R 0 We say that u and V are harmonic conjugates of one another 0 Given an harmonic function u one can use the Cauchy Riemann equations to find its harmonic conjugate V and viceversa 0 Examples 0 Check that uxy 2xy is harmonic and find its harmonic conjugate V 0 Given an harmonic function VXy how would you find its harmonic conjugate uxy Chapter 13 Complex Numbers

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