We shall see for example in theorems 54 and 55 that

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We shall see (for example in Theorems 5.4 and 5.5) that the study of power series and complex differentiable functions are closely linked. In Analysis I you studied a particular power series of great importance to us. Theorem 1.4. (i) n =0 z n /n ! converges for all z . If we write exp z = summationdisplay n =0 1 n ! z n , then exp is everywhere differentiable with exp z = exp z . (ii) exp z exp w = exp( z + w ) for all z, w C . (iii) The equation exp z = 0 has no solution. If w negationslash = 0 the equation exp z = w has the solutions z = log | w | + + 2 nπi with n Z , where θ is any particular real solution of w | w | = cos θ + i sin θ. (iv) If we write e z = exp z and sin z = e iz e iz 2 i , cos z = e iz + e iz 2 then, when z is real, we recover the traditional real functions sin : R R and cos : R R . Combining the results of this section, we see that we have obtained a useful library of complex differentiable functions. 2 Complex differentiability is not like real dif- ferentiability The first hint that complex differentiability is different from real differentia- bility is given by the following example. 3
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Example 2.1. The function F : C C given by F ( z ) = z is nowhere differentiable. To understand Example 2.1 it is helpful to view matters not algebraicly (as we did in Section 1) but geometrically. Observe that, if we ignore multi- plication, C can be considered as the vector space R 2 . If we have a function f : C C we can write f ( x + iy ) = u ( x,y ) + iv ( x,y ) with x , y , u and v real, obtaining the map from R 2 R 2 parenleftbigg x y parenrightbigg mapsto→ parenleftbigg u ( x,y ) v ( x,y ) parenrightbigg . Theorem 2.2. If the map T : R 2 R 2 given by parenleftbigg x y parenrightbigg mapsto→ parenleftbigg u ( x,y ) v ( x,y ) parenrightbigg is differentiable in the sense of the course Analysis II (Course P9), then the following statements are equivalent. (i) f is complex differentiable at z 0 . (ii) The map h mapsto→ f ( z 0 + h ) f ( z 0 ) is locally the composition of a rotation and a dilation. (iii) The Jacobian matrix of the map T satisfies parenleftbigg ∂u ∂x ∂u ∂y ∂v ∂x ∂v ∂y parenrightbigg = λ parenleftbigg cos θ sin θ sin θ cos θ parenrightbigg with λ real and λ 0 . (iv) The map T satisfies the Cauchy-Riemann conditions ∂u ∂x = ∂v ∂y , ∂v ∂x = ∂u ∂y . Thus z mapsto→ z is not complex differentiable because it is a reflection. Most Cambridge examinees and a worryingly high proportion of Cam- bridge examiners believe that the Cauchy-Riemann relations are the best way of testing for complex differentiability. This is not the case in general. The methods of Section 1 usually furnish a more efficient tool. (One problem with the use of the Cauchy-Riemann equations is that, as is shown in Anal- ysis II, the existence of partial derivatives does not imply differentiability.) In any case it is the act of a lunatic (or a Cambridge examiner) to ask about complex differentiability at a single point. The subject of complex differentiability only becomes interesting when applied to functions differen- tiable on an open set.
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