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Unformatted text preview: Informal lecture notes for Nov. 29 1 Complex differentiation, continued 1.1 Complex differentiation rules Simple functions like constant functions and f ( z ) = z can be differentiated just using the definition, as we saw. However, just as in the real case, for more complicated functions we rely on various differentiation rules. The following rules extend the familiar rules from real calculus to the complex setting. In fact, they are proved in essentially the same way, so we wont give the proofs here. Theorem: Let f ( z ) and g ( z ) be holomorphic functions. Then the following are true: 1. (Linearity) For any constants c, d C , cf ( z ) + dg ( z ) is holomorphic and ( cf ( z ) + dg ( z )) = cf ( z ) + dg ( z ). 2. (Product rule) The product ( fg )( z ) is holomorphic and ( fg ) ( z ) = f ( z ) g ( z )+ f ( z ) g ( z ). 3. (Quotient rule) The quotient f ( z ) /g ( z ) is holomorphic wherever g ( z ) 6 = 0 and f ( z ) g ( z ) = f ( z ) g ( z )- f ( z ) g ( z ) g 2 ( z ) . 4. (Chain rule) The composite function f ( g ( z )) is holomorphic whenever g ( z ) is in the domain of f ( z ), and ( f ( g )) ( z ) = f ( g ( z )) g ( z ). Of course, these rules are are only useful if we already have functions f and g , the derivatives of which we know. In the last lecture, we showed that the derivative of any constant function is 0 everywhere and the derivative of f ( z ) = z is 1 everywhere. Using this and the product rule, we see that ( z 2 ) = ( z z ) = 1 z + z 1 = 2 z. Iterating this procedure, we see that ( z n ) = nz n- 1 . Finally, using the above and linearity, we see that any (complex) polynomial, which we write f ( z ) = c n z n + c n- 1 z n- 1 + + c 1 z + c , is holomorphic on all of C and has derivative f ( z ) = nc n z n- 1 + ( n- 1) c n- 1 z n- 2 + 2 c 2 z + c 1 . 1 Going beyond polynomials, observe that, using the quotient rule, we can write 1 z n = z n- 1 nz n- 1 z 2 n =- n z n +1 . (On the homework, you are asked to show that (1 /z ) =- 1 /z 2 by a different method.) More generally, the quotient rule and the above formula for polyno- mials allows us to differentiate any rational function. These computations for polynomials and rational functions are a good start. However, its not immediately clear what to do about differentiating e z , for example. Moreover, we would like to go beyond elementary functions and look at general complex functions f ( x + iy ) = u ( x, y ) + iv ( x, y ). How to do this is the subject of the next section. 1.2 The Cauchy-Riemann equations Consider a general complex function f ( x + iy ) = u ( x, y ) + iv ( x, y ). As we saw last lecture, a necessary condition for it to be complex differentiable is that the difference quotient gives the same limit along both the real and imaginary axis (this is how we showed that Re( z ) is not holomorphic). Assume that f ( z ) = f ( x + iy ) is holomorphic. Then it must be true that, for real t , f ( z ) = lim t f ( z + t )- f ( z ) t = lim...
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- Spring '07