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5 Homework 5 1. Show that 2 sinz · sin w = cos(z − w) − cos(z + w) for any z, w ∈ C. 2. Find power series expansion of f(z) = log(4 + 3z) at the point z = −1. 3. If α > 1, show: Q∞ n=1 1 − z nα converges uniformly on compact subsets of C. 4. The famous Fiboniacci numbers is a sequence defined by a0 = a1 = 1, an = an−1+an−2 for n ≥ 2. Compare this sequence with the coefficients of the power series centered at 0 of the function 1 z 2+z−1 . 6 Homework 6 1. Find the image of the right half-plane Re(z) > 0 under the linear transformation w = f(z) = i(1−z) 1+z . 2. Find the linear transformation w = f(z) that maps the points z1 = −i, z2 = 0, z3 = i onto w1 = −1, w2 = i and w3 = 1 respectively. 3. Find the fixed points of w = 4z+3 2z−1 (by fixed point, we mean the point z0 such that f(z0) = z0). 4. Find a linear transformation w = f(z) such that it maps the disk ∆(2) onto the right half-plane {w | Re(w) > 0} satisfying f(0) = 1 and arg f′ (0) = π 2 . 7 Homework 7 1. Show: f(z) = x 2 + iy2 satisfies the Cauchy-Riemann equation at the origin, but is not holomorphic in any neighborhood of 0. 2. Show: f(z) = |z| 2 + z 2 is not a holomorphic function. 3. Let f(z) = x 2 + axy + by2 + i(cx2 + dxy + y 2 ). What kind of values for the constants a, b, c, d, should we take so that f is holomorphic on C? 4. If f(z) is a holomorphic function with u(x, y) = 2(x − 1)y, f(2) = −i. Find f(z) by consideration of the Cuachy-Riemann equation. 5. If f(z) = u+iv is a holomorphic function on a domain D satisfying |f(z)| = constant, show that f(z) is constant.

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