ELEC3002-complex2 - ELEC3002/ELEC7005 Lecture 2 Complex...

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ITEE complex 2/1 ELEC3002/ELEC7005 ELEC3002/ELEC7005 Lecture 2 Complex Analysis (chapter 1 in Glyn James) complex differentiation This lecture covers sections 1.3 and 1.4
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ITEE complex 2/2 Complex differentiation Complex differentiation The derivative of a real function is straight-forward. ( 29 ( 29 ( 29 0 0 0 0 lim x x f x f x f x x x f - p = - how about we replace x by z ? However, note that the complex variable is a function of both x & y and the limit can be approached along different directions. In the end it turns out that we have to look at different differentiability conditions know as the Cauchy Riemann conditions to determine whether the derivative exists.
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ITEE complex 2/3 A few definitions (real important) A few definitions (real important) A function is said to be analytic (equivalently holomorphic or regular) in a domain if the function is both defined and is differentiable in that domain. Example: Polynomials are analytic in the entire complex plane, but rational functions of the form f(z) = g(z)/h(z) are analytic at all points except when h(z) =0 Cauchy –Riemann equations: ; u v u v x y y x f = = - These equations are really a test for analyticity of a complex function. If a function satisfies these equations, it is analytic. (see proof in text )
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ITEE complex 2/4 Example Example ( 29 2 f z z = analytic? ( 29 2 2 2 2 2 2 2 ; 2 w z x jy x y jxy u x y v xy = = + = - + = - = 2 ; 2 ; 2 ; 2 u u v v x y y x x y x y f = = - = = so Cauchy Riemann equations have: ; u v u v x y y x f = = - Analytic! in fact, it is an entire function (analytic everywhere)
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ITEE complex 2/5 CR-in polar form CR-in polar form What about the function: ( 29 ? w f z z = = Sometimes we use the polar form of the Cauchy Riemann conditions: If ( 29 cos sin z r j θ θ = + and if we set ( 29 ( 29 ( 29 , , f z u r jv r θ θ = + Then 1 1 ; u v v u r r r r θ θ = = -
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ITEE complex 2/6 conjugate functions conjugate functions A pair of functions u(x,y) and v(x,y) that satisfy the CR conditions are called conjugate functions (note: this is different from the conjugate complex number just considered). Conjugate functions are orthogonal to each other, in that curves in the (x,y ) plane for u(x,y) = const are orthogonal to curves v(x,y) = cons t.
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