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TAKE HOME FINAL EXAM FOR MATH 552 Instructions: Each problem is worth the same amount. Write all answers and show all work in a blue book. Due May 1 at 9am. Problem #1: Define the set C of complex numbers in terms µ of 2 ×¶2 0 0 real matrices. Show using this definition that if z ∈ C and z 6= 0 0 ¶ µ 1 0 . Explain what we then there exists z −1 ∈ C such that z z −1 = 0 1 mean when we write z = x + iy, where x, y ∈ R and i2 = −1. Problem #2: Assume that the Ratio Test is applicable for series P∞ n=0 zn with zn ∈ C, zn 6= 0, n ≥ 0, i.e. the series converges (abso| lutely) if limn→∞ |z|zn+1 exists and is less than 1. n| P∞ zn • Show that n=0 n! converges for every z ∈ C. • Argue from the Converse of Taylor’s Theorem that the sum of this series defines an entire function, and denote it by ez . • Show that ez+w = ez ew . P zn Note: At the beginning of the semester we defined ez = ∞ n=0 n! and put off questions of convergence until later. Now we see that our procedure was justified. Problem #3: Assuming the series expansions cos z = P Pusual ∞ ∞ n z 2n n z 2n+1 n=0 (−1) (2n)! and sin z = n=0 (−1) (2n+1)! hold for all z ∈ C show that Euler’s formula eiθ = cos θ + i sin θ is true for all θ ∈ R. Problem #4: • Show that the triangle inequality |z + w| ≤ |z| + |w| holds for all z, w ∈ C. • Define what it means to assert: limz→z0 f (z) = L. • Show that if z0 is an accumulation point of the domain set of the function f and limz→z0 f (z) = L1 and limz→z0 f (z) = L2 then L1 = L2 . 1 2 Problem #5: Assume U ⊂ C is open and z0 = x0 + iy0 ∈ U and f : U → C is a function. • Define (in terms of limits) what it means to assert f is Cdifferentiable at z0 . • If u(x, y) = ℜf (x + iy) and v(x, y) = ℑf (x + iy) whenever x, y ∈ R, x + iy ∈ U , then show that the Cauchy-Riemann equations hold at (x0 , y0 ) whenever f is C-differentiable at z0 . • Suppose the Cauchy-Riemann equations hold at (x0 , y0 ). Give additional conditions on u and v which are sufficient to guarantee that f is C-differentiable at z0 . Problem #6: • For nonzero z ∈ C define Ln(z). • Show that Ln(z) is analytic on the set C \ (−∞, 0]. d • Show that dz Ln(z) = z1 for z ∈ C \ (−∞, 0]. Problem #7: Suppose Z : [a, b] → C is a parameterization of a smooth arc C = range(Z), and f : C → C is a function so that f (Z(t)) is continuous for t ∈ [a, b]. R • Define the contour integral C f (z) dz. • Give a parameterization forR the reversed contour −C. R • Show that −C f (z) dz = − C f (z) dz. Problem #8: • Define what it means to assert that D ⊂ C is a domain. • Suppose D is a domain and f is analytic on D and real-valued on D. Show that f (z) is constant. Problem #9: Suppose D is a domain and f (z) is continuous on D. • Does an antiderivative F (z) necessarily exist for f (z)? Hint: consider f R(z) = z1 for z ∈ D = C \ {0}. • Compute C z n dz, for each n ∈ Z, where C is the unit circle oriented in the counterclockwise direction. Note: Antiderivatives always exist for continuous real-valued functions defined on intervals (a, b) ⊂ R. This is a consequence of the Fundamental Theorem of Calculus. Problem #10: Prove the Cauchy-Goursat Theorem for a rectangle (without assuming f ′ (z) is continuous). You should use but do not need to prove the following lemma. 3 Lemma. Suppose R = [a, b] × [c, d] and f (z) is analytic at each point of R. For every ǫ > 0 there exists an integer n ≥ 1, determining a subdivision of R into n2 subrectangles R1 , . . . , Rn2 each of which are of dimensions (b−a) × (d−c) , and there exist points zj ∈ Rj , 1 ≤ j ≤ n2 n n such that for every 1 ≤ j ≤ n2 and every point z 6= zj on the boundary of Rj we have that ¯ ¯ ¯ f (z) − f (zj ) ¯ ¯ ¯ < ǫ. − f (z ) j ¯ z − zj ¯ Problem #11: • State the Cauchy Integral Formula for f (n) (z0 ) with all its needed hypotheses. • Assuming this result is true show that the case n = 2 allows us to conclude that derivatives of analytic functions are themselves analytic. • If f is an analytic function and f = u + iv as in problem #5 then show that u and v have continuous partial derivatives of all orders. Note: The function f (x) = x|x| is R-differentiable at all x ∈ R and f ′ (x) = |x|. However f ′ is not R-differentiable at x = 0. Problem #12: State and prove Morera’s Theorem using the second part of problem #11. Problem #13: State and prove Taylor’s Theorem. Problem #14: State a converse to Taylor’s Theorem. Problem #15: State Laurent’s Theorem. Problem #16: Use the Cauchy Residue Theorem to show that π . = 2√ 2 R∞ 0 dx x4 +1