Math 185 Homework Exercises II

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1 Math 185 Homework Exercises II Instructor: Andrés E. Caicedo Due: July 10, Verify that if f H(Ω) C 2 (Ω) is never zero, then ln f is harmonic in Ω. 2. Let f = u+iv H(Ω) C 2 (Ω). Let p 2 be an integer. Show the following identities (be careful about the points where f(z) = 0 or u(z) = 0): (a) f(z) p = p 2 f(z) p 2 f (z) 2 ; (b) u(z) p = p(p 1) u(z) p 2 f (z) 2 ; (c) z u(z) = 1 2 f (z) and z v(z) = 1 2i f (z); ( (d) e p f(z) = pe p f(z) p + 1 ) f (z) 2 (f(z) 0). f(z) 3. Consider the transformation z e z : (a) What is the image of a family of lines parallel to one of the coordinate axes? (The answer depends on the axis.) (b) What is the image of a line not parallel to any of the coordinate axes? 4. Let f(z) = u(x, y) + iv(x, y) be holomorphic. (a) Write down the Cauchy-Riemann equations in polar coordinates. That is, show that if (r, θ) are the corresponding polar coordinates of z (i.e., z = re iθ, 0 < r), then the Cauchy-Riemann equations become u r = 1 v r θ, v r = 1 u r θ. (b) Let {s, n} be an orthonormal basis of R 2 defining the usual orientation (i.e., n = is and n = s = 1). Let n and s represent differentiation directions n and s, respectively so, for example, and similarly for s f(z 0) = n. lim h 0 h R 1 f(z 0 + hs) f(z 0 ), h

2 Then the Cauchy-Riemann equations become u s = v n, u n = v s. 5. From Churchill & Brown, Complex Variables and Applications, 5th edn., In detail: , 4, , , 2.a,b,c, 5, , 6, 7, 8.a,b,d, 12, 13, 14, Write the function f(z) = z 3 + z + 1 in the form f(z) = u(x, y) + iv(x, y). 4. Sketch the region onto which the sector r 1, 0 θ π/4 is mapped by the transformation (a) w = z 2 ; (b) w = z 3 ; (c) w = z Another interpretation of a function w = f(z) = u(x, y) + iv(x, y) is that of a vector field in the domain of definition of f. The function assigns a vector w, with components u(x, y) and v(x, y), to each point z at which it is defined. Indicate graphically the vector fields represented by the equations (a) w = iz; (b) w = z/ z Show that lim z z0 f(z)g(z) = 0 if lim z z0 f(z) = 0 and if there exists a positive number M such that g(z) M for all z in some neighborhood of z Show that a set S C is unbounded if and only if every neighborhood of the point at infinity contains at least one point in S. 2

3 Let f denote the function whose values are ( z) 2 when z 0, f(z) = z 0 when z = 0. Show that if z = 0 then w/ z = 1 at each nonzero point on the real and imaginary axes in the z plane and that w/ z = 1 at each nonzero point ( x, x) on the line y = x in that plane, where z = ( x, y). Conclude from these observations that f (0) does not exist Use the theorem in Sec. 17 (If f (z) exists then the Cauchy-Riemann equations hold at z and f (z) = x f(z)) to show that f (z) does not exist at any point if (a) f(z) = z; (b) f(z) = z z; (c) f(z) = 2x + ixy 2 ; (d) f(z) = e x e iy. 2. Use the theorem in Sec. 18 (Let f be defined in some neighborhood of z 0. Suppose the first-order partial derivatives of f with respect to x and y exist everywhere in that neighborhood and that they are continuous at z 0. Then, if the Cauchy-Riemann equations hold at z 0, the derivative f (z 0 ) exists) to show that f (z) and its derivative f (z) exist everywhere, and find f (z) when (a) f(z) = iz + 2; (b) f(z) = e x e iy ; (c) f(z) = z Show that when f(z) = x 3 + i(1 y) 3, it is legitimate to write f (z) = u x + iv x = 3x 2 only when z = i. 6. Let u and v denote the real and imaginary components of the function f defined by the equations ( z) 2 when z 0, f(z) = z 0 when z = 0. Verify that the Cauchy-Riemann equations u x = v y and u y = v x are satisfied at the origin z = (0, 0). 3

4 With the aid of the theorem in Sec. 17, show that each of these functions is nowhere analytic: (a) f(z) = xy + iy; (b) f(z) = e y e ix. 6. Use results in Sec. 19 (If f is defined in some neighborhood of z 0, if the first order partial derivatives of f with respect to r and θ exist everywhere in that neighborhood and are continuous at z 0, and if the polar form of the Cauchy-Riemann equations hold at z 0, then the derivative f (z 0 ) exists) to verify that the function g(z) = ln r + iθ (r > 0, 0 < θ < 2π) is analytic in the indicated domain of definition, with derivative g (z) = 1/z. Then show that the composite function g(z 2 + 1) is an analytic function of z in the quadrant x > 0, y > 0, with derivative 2z/(z 2 + 1). Suggestion: Observe that I(z 2 + 1) > 0 when x > 0, y > Let a function f(z) be analytic in a domain Ω. Prove that f(z) must be constant in Ω if either of the following holds: (a) f(z) is real-valued for all z Ω; (b) f(z) is analytic in Ω; (c) f(z) is constant in Ω. Suggestion: Use the Cauchy-Riemann equations to prove parts (a) and (b). To prove part (c), observe that either f(z) = 0, and we are done, or f(z) = c 2 /f(z) if f(z) = c 2 0. Then use part (b). 8. Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) when (a) u(x, y) = 2x(1 y); (b) u(x, y) = 2x x 3 + 3xy 2 ; (d) u(x, y) = y/(x 2 + y 2 ). 12. Let the function f(z) = u(x, y) + iv(x, y) be analytic in a domain Ω, and consider the families of level curves u(x, y) = c 1 and v(x, y) = c 2, where c 1 and c 2 are arbitrary real constants. Prove that these families are orthogonal. More precisely, show that if z 0 = x 0 + iy 0 Ω is a point common to two particular curves u(x, y) = c 1 and v(x, y) = c 2 and if f (z 0 ) 0, then the lines tangent to those curves at (x 0, y 0 ) are perpendicular. 4

5 Suggestion: Note how it follows from the equations u(x, y) = c 1 and v(x, y) = c 2 that u x + u dy y dx = 0 and v x + v dy y dx = Show that when f(z) = z 2, the level curves u(x, y) = c 1 and v(x, y) = c 2 of the component functions are hyperbolas asymptotic to the coordinate axes, or to the lines y = x and y = x. Note the orthogonality of the two families, described in Exercise 12. Observe that the curves u(x, y) = 0 and v(x, y) = 0 intersect at the origin but are not, however, orthogonal to each other. Why is this fact in agreement with the result in Exercise 12? 14. Sketch the families of level curves of the component functions u and v when f(z) = 1/z, and note the orthogonality described in Exercise Sketch the families of level curves of the component functions u and v when f(z) = z 1 z + 1, and note how the result in Exercise 12 is illustrated here. 5

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