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# Class Note for MATH 322 with Professor Glickenstein at UA

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This 15 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 18 views.

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Date Created: 02/06/15

Limits continuity and differentiation A criterion for analyticity quot Con rm12x Wimmbgm Function of a complex variable Limits continuity and differentiation Limits and continuity A criterion for analytici39iy Differentiability Analytic functions 1 Function of a complex variable o A single valued function f of a complex variable 2 is such that for every 2 in the domain of definition D of 7 there is a unique complex number W such that W fz o The real and imaginary parts of 2 often denoted by u and v are such that fzuxyivgtlty7 ZXiy7 X7yER fZ C with uxy E R and vx y E R Chapter 13 Complex Numbers Function of a complex variable Limits continuity and differentiation Limits and continuity A criterion for analyticitjv39 Differentiabili39ty Analytic functions Function of a complex variable continued 0 Examples 0 fz z is such that ux y x and vx y y 0 Find the real and imaginary parts of fz 2 1 o fz E is defined for all z y 0 and is such that X y XayX2y27 VX7yX2y239 Chapter 13 Complex Numbers Function of a complex variable Limits continuity and differentiation Limits and continuity A criterion for analyticity 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 zo lt e 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 Le that lim fz W0 z gtzo if for every 6 gt 0 one can find 6 gt 0 such that for all z E D z 20 lt6gt fz W0 lt6 2 1 0 Example lim Z 22 z gti Z i Chapter 13 Complex Numbers Function of a complex variable Limits continuity and differentiation Limits and continuity A criterion for analyticiw Differentiabili39ty Analytic functions Con nuhy o The function f is continuous at z 20 if f is defined in a neighborhood of 20 including at z 20 and lim fz fzo z gtzo o If fz is continuous at z 20 so is fz Therefore if f is continuous at z 20 so are 32 mf and f2 0 Conversely if uX y and vX y are continuous at X0y0 then fz uxy ivX y with z X iy is continuous at 20 2 X0 l iyo 0 Example Is the function such that fz mz2z2 for z 75 O and f0 0 continuous at z 0 Chapter 13 Complex Numbers Function of a complex variable Limits continuity and differentiation Limits and continuity A criterion for analytici39iy Differentiability Analytic functions 3 Differtiability 0 Assume that f is defined in a neighborhood of z 20 The derivative of the function f at z 20 is fZO m fzo l AZ fzo m fz fzo7 Az gt0 AZ z gtzo z 20 assuming that this limit exists 0 If f has a derivative at z 20 we say that f is differentiable at Z 20 0 Examples 0 fz 2 is continuous but not differentiable at z 0 o fz Z3 is differentiable at any 2 E C and f z 322 Chapter 13 Complex Numbers Function of a complex variable Limits continuity and differentiation Limits and continuity A criterion for analyticitjv39 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 Z2 d o ZnnZn 1 dz 7 c Find lim 2 z gt I Z Chapter 13 Complex Numbers Function of a complex variable Limits continuity and differentiation Limits and continuity A criterion for analyticity Differentiability Analytic functions Rules forontinuity 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 a The limit of a product sum is the product sum of the limits 0 The product and quotient rules for differentiation still apply 0 The chain rule still applies 0 Examples 2 0 Find 12 1 dz z i d 3 4 c Find Ez 9z 7 Chapter 13 Complex Numbers Function of a complex variable Limits continuity and differentiation Limits and continuity A criterion for analyticiw Differentiabili39ty Analytic functions 4 Analyic functions 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 Function of a complex variable Limits continuity and differentiation Limits and continuity A criterion for analytici39iy Differentiabili39ty Analytic functions Analytic nctions 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 fz 32 722 z3 is analytic at every 2 o Rational functions of a complex variable of the form 2 fz where g and h are polynomials are analytic z everywhere except at the zeros of hz z2 l 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 Cauchy Riemann equations A criterion for analyticity Harmonic functions 5 The Cuchy Riemann equations 0 If f2 uxy ivX y is defined in a neighborhood of z Xl iy and if f is differentiable at 2 then UXX7y VyX7Y7 and uyX7 VXX7y These are called the Cauchy Riemann 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 Cauchy Riemann equations at 2 then fz uX y l ivXy is differentiable at z o The Cauchy Riemann equations therefore give a criterion for analyticity Chapter 13 Complex Numbers Limits continuity and differentiation The Cauchy Riemann equations A criterion for analyticity Harmonic functions The Cau Riemann equations coinued 0 Indeed if a function is analytic at 2 it must satisfy the Cauchy Riemann equations in a neighborhood of z In particular if 7 does not satisfy the Cauchy Riemann equations then 7 cannot be analytic 0 Conversely if the partial derivatives of u and v exist are continuous and satisfy the Cauchy Riemann equations in a neighborhood of z X l iy then fz uxy l ivx y is analytic at z 0 Examples 0 Use the Cauchy Riemann equations to show that 2 is not analytic 1 0 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 Cauchy Riemann equations A criterion for analyticity Harmonic functions Applicatins of the Cauchy Riemannquations o A consequence of the Cauchy Riemann equations is that fz uX l ivX vy iuy 2 0 We will use these formulas later to calculate the derivative of some analytic functions 0 Another consequence of the Cauchy Riemann 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 differentiation The Cauchy Riemann 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 fz uXy ivX y is analytic in R then both u and v satisfy Laplace39s equation in R ie V2uuxxuyy0 and V2v2vXX l vyy20 3 o A function that satisfies Laplace39s 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 fz uxy ivX y 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 vice versa 0 Examples 0 Check that uX y 2xy is harmonic and find its harmonic conjugate v 0 Given an harmonic function vX y how would you find its harmonic conjugate uX y Chapter 13 Complex Numbers

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