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Exercise 1. Analytic functions [10 points] Investigate using the Cauchy-Riemann equations which of the following functions are analytic in z = r + iy, and if so, in which domain in C. Check your conclusions by expressing each function in terms of z and z*. fi(x,y) = x+x- y + zy(2x + 1) f2 (x, y) = ry +is Hint: recall that r = (z + 2*)/2 and y = (z - z*) /2i. Exercise 2. Laurent expansion [10 points] Using the known Taylor expansion of es obtain the Laurent series expansion of Option 1: basic problem ez /2 about z = 0 Option 2: advanced problem zez /(z - 1) about z = 1. Exercise 3. Computing residues [10 points] Compute the residue of the following function at its pole: 223 - 322 f ( = ) = (2 - 2)3 Exercise 4. Residue theorem [10 points] Compute the following integral with the residue theorem: sin & dz. 121=6 22 - 4 Hint: when solving problems of this kind it is helpful to make a sketch showing the poles of the integrand and the integration contour