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Example text
3-6) or Eq. 3-8). These ideas are illustrated in the following. 4 The Derivative and Analyticity the point z = o will contain points at which f f ( z ) fails to exist. Hence, f ( z ) is not analytic at z = 0 (or anywhere else). EXAMPLE 4 For what values of z is the function f ( z ) = x2 au -=2x=2y=ax av ay and av -=O=--. ax any ,-onstant, is the product of entire functions and is also entire. A polyno. ao is entire since it is a sum of entire expression a,r " a , - ~ z + "-'+ + + iy2 analytic? From the C-R equations, with u = x2, v = y2, we have Solution.
The maximum value achieved by I f (z) I in R will occur when ex is maximum, that for example: Reinhold Remmert, Theoq of Complex Functions (New York: Springer-Verlag, 19901, C h a ~ k r0, section 5. This same section contains historical information on the concepts of functions and COntinuim. 'continuity for u(x, Y), a rpul function of two real variables, is defined in a way analogous to continuity for can be made smaller than any positive all (x, y) lying inside a circle of radius 6 centered at (xo, YO).
Give the numerical value. b) Where is this function analytic? 77 + ( x - 1)' + ixy at any points where the derivative + 3. a) Where is the function f ( z ) = z 3 + z2 1 analytic? b) Find an expression for f f ( z ) and give the derivative at 1 + i. have a derivative? 4. a) W e r e does the function f ( z ) = z2 + (x - 1)2 + i ( y The equations can be used for all r, 8 except where r = 0. I n the same problem, we show that if the derivative of f ( z ) exists it can be found from either of the following: b) Where is this hnction analytic?