Multivariable Advanced Calculus

10.8. EXERCISES 291for all x ∈ ∂Ω. Show that thend (f − g, Ω, 0) = d (f, Ω, 0)Show that if there exists x ∈ f −1 (0) , then there exists x ∈ (f − g) −1 (0). Hint:You might consider h (t, x) ≡ (1 − t) f (x)+t (f (x) − g (x)) and argue 0 /∈ h (t, ∂Ω)for t ∈ [0, 1].18. Let f : C → C where C is the field of complex numbers. Thus f has a real andimaginary part. Letting z = x + iy,f (z) = u (x, y) + iv (x, y)Recall that the norm in C is given by |x + iy| = √ x 2 + y 2 and this is the usualnorm in R 2 for the ordered pair (x, y) . Thus complex valued functions defined onC can be considered as R 2 valued functions defined on some subset of R 2 . Such acomplex function is said to be analytic if the usual definition holds. That isIn other words,f ′ f (z + h) − f (z)(z) = lim.h→0 hf (z + h) = f (z) + f ′ (z) h + o (h) (10.26)at a point z where the derivative exists. Let f (z) = z n where n is a positiveinteger. Thus z n = p (x, y) + iq (x, y) for p, q suitable polynomials in x and y.Show this function is analytic. Next show that for an analytic function and u andv the real and imaginary parts, the Cauchy Riemann equations hold.u x = v y , u y = −v x .In terms of mappings show 10.26 has the form( u (x + h1 , y + h 2 )v (x + h 1 , y + h 2 )==( ) u (x, y)v (x, y)(u (x, y)v (x, y))(ux (x, y) u+y (x, y)v x (x, y) v y (x, y))+(ux (x, y) −v x (x, y)v x (x, y) u x (x, y)) ( )h1+ o (h)h 2) ( )h1+ o (h)h 2where h = (h 1 , h 2 ) T and h is given by h 1 +ih 2 . Thus the determinant of the abovematrix is always nonnegative. Letting B r denote the ball B (0, r) = B ((0, 0) , r)showd (f, B r , 0) = n.where f (z) = z n . In terms of mappings on R 2 ,( u (x, y)f (x, y) =v (x, y)Thus showHint: You might considerd (f, B r , 0) = n.g (z) ≡n∏(z − a j )j=1where the a j are small real distinct numbers and argue that both this functionand f are analytic but that 0 is a regular value for g although it is not so for f.However, for each a j small but distinct d (f, B r , 0) = d (g, B r , 0).).