Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX VARIABLESwhere i and j are the unit vectors along the x- andy-axes, respectively. A similarexpression exists for ∇v, the normal to the curve v(x, y) = constant. Taking thescalar product of these two normal vectors, we obtain∇u · ∇v = ∂u ∂v∂x ∂x + ∂u ∂v∂y ∂y= − ∂u ∂u∂x ∂y + ∂u ∂u∂y ∂x =0,where in the last line we have used the Cauchy–Riemann relations to rewritethe partial derivatives of v as partial derivatives of u. Since the scalar productof the normal vectors is zero, they must be orthogonal, and the curves u(x, y) =constant and v(x, y) = constant must therefore intersect at right angles.◮Use the Cauchy–Riemann relations to show that, for any analytic function f = u + iv, therelation |∇u| = |∇v| must hold.From (24.9) we have( ) 2 ( ) 2 ∂u ∂u|∇u| 2 = ∇u · ∇u = + .∂x ∂yUsing the Cauchy–Riemann relations to write the partial derivatives of u in terms of thoseof v, weobtain( ) 2 ( ) 2 ∂v ∂v|∇u| 2 = + = |∇v| 2 ,∂y ∂xfrom which the result |∇u| = |∇v| follows immediately. ◭24.3 Power series in a complex variableThe theory of power series in a real variable was considered in chapter 4, whichalso contained a brief discussion of the natural extension of this theory to a seriessuch as∞∑f(z) = a n z n , (24.10)n=0where z is a complex variable and the a n are, in general, complex. We nowconsider complex power series in more detail.Expression (24.10) is a power series about the origin and may be used forgeneral discussion, since a power series about any other point z 0 can be obtainedby a change of variable from z to z − z 0 .Ifz were written in its modulus andargument form, z = r exp iθ, expression (24.10) would becomef(z) =∞∑a n r n exp(inθ). (24.11)n=0830