Math 185 — Homework Exercises II

Instructor: Andres E. Caicedo

Due: July 10, 2002

1. Verify that if f ∈ H(Ω)∩C2(Ω) is never zero, then ln |f | is harmonic in Ω.

2. Let f = u+iv ∈ H(Ω)∩C2(Ω). Let p ≥ 2 be an integer. Show the followingidentities (be careful about the points where f(z) = 0 or u(z) = 0):

(a) ∆|f(z)|p = p2|f(z)|p−2|f ′(z)|2;(b) ∆|u(z)|p = p(p− 1)|u(z)|p−2|f ′(z)|2;(c)

∂

∂zu(z) =

12f ′(z) and

∂

∂zv(z) =

12i

f ′(z);

(d) ∆ep|f(z)| = pep|f(z)|(

p +1

|f(z)|)|f ′(z)|2 (f(z) 6= 0).

3. Consider the transformation z 7→ ez:

(a) What is the image of a family of lines parallel to one of the coordinateaxes? (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. Thatis, show that if (r, θ) are the corresponding polar coordinates of z (i.e.,z = reiθ, 0 < r), then the Cauchy-Riemann equations become

∂u

∂r=

1r

∂v

∂θ,

∂v

∂r= −1

r

∂u

∂θ.

(b) Let s, n be an orthonormal basis of R2 defining the usual orien-

tation (i.e., n = is and |n| = |s| = 1). Let∂

∂nand

∂

∂srepresent

differentiation directions n and s, respectively so, for example,

∂

∂sf(z0) = lim

h → 0h ∈ R

f(z0 + hs)− f(z0)h

,

and similarly for∂

∂n.

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Then the Cauchy-Riemann equations become

∂u

∂s=

∂v

∂n,

∂u

∂n= −∂v

∂s.

5. From Churchill & Brown, Complex Variables and Applications, 5th edn.,

• §2.10. 2, 4, 10

• §2.14. 5, 13

• §2.16. 7

• §2.19. 1, 2.a,b,c, 5, 6

• §2.21. 2, 6, 7, 8.a,b,d, 12, 13, 14, 16

In detail:

1 §2.10.

2. Write the function f(z) = z3 + 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 bythe transformation

(a) w = z2;

(b) w = z3;

(c) w = z4.

10. Another interpretation of a function w = f(z) = u(x, y) + iv(x, y) is thatof a vector field in the domain of definition of f . The function assigns avector w, with components u(x, y) and v(x, y), to each point z at whichit is defined. Indicate graphically the vector fields represented by theequations

(a) w = iz;

(b) w = z/|z|.

2 §2.14.

5. Show that limz→z0 f(z)g(z) = 0 if limz→z0 f(z) = 0 and if there exists apositive number M such that |g(z)| ≤ M for all z in some neighborhoodof z0.

13. Show that a set S ⊆ C is unbounded if and only if every neighborhood ofthe point at infinity contains at least one point in S.

2

3 §2. 16.

7. Let f denote the function whose values are

f(z) =

(z)2

zwhen z 6= 0,

0 when z = 0.

Show that if z = 0 then ∆w/∆z = 1 at each nonzero point on the real andimaginary axes in the ∆z plane and that ∆w/∆z = −1 at each nonzeropoint (∆x,∆x) on the line ∆y = ∆x in that plane, where ∆z = (∆x, ∆y).Conclude from these observations that f ′(0) does not exist.

4 §2. 19.

1. Use the theorem in Sec. 17 (If f ′(z) exists then the Cauchy-Riemann

equations hold at z and f ′(z) =∂

∂xf(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 + ixy2;(d) f(z) = exe−iy.

2. Use the theorem in Sec. 18 (Let f be defined in some neighborhood ofz0. Suppose the first-order partial derivatives of f with respect to x and yexist everywhere in that neighborhood and that they are continuous at z0.Then, if the Cauchy-Riemann equations hold at z0, the derivative f ′(z0)exists) to show that f ′(z) and its derivative f ′′(z) exist everywhere, andfind f ′′(z) when

(a) f(z) = iz + 2;(b) f(z) = e−xe−iy;(c) f(z) = z3.

5. Show that when f(z) = x3 + i(1 − y)3, it is legitimate to write f ′(z) =ux + ivx = 3x2 only when z = i.

6. Let u and v denote the real and imaginary components of the function fdefined by the equations

f(z) =

(z)2

zwhen z 6= 0,

0 when z = 0.

Verify that the Cauchy-Riemann equations ux = vy and uy = −vx aresatisfied at the origin z = (0, 0).

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5 §2.21.

2. With the aid of the theorem in Sec. 17, show that each of these functionsis nowhere analytic:

(a) f(z) = xy + iy;

(b) f(z) = eyeix.

6. Use results in Sec. 19 (If f is defined in some neighborhood of z0, if thefirst order partial derivatives of f with respect to r and θ exist everywherein that neighborhood and are continuous at z0, and if the polar form of theCauchy-Riemann equations hold at z0, then the derivative f ′(z0) 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(z2 + 1) is an analyticfunction of z in the quadrant x > 0, y > 0, with derivative 2z/(z2 + 1).

Suggestion: Observe that =(z2 + 1) > 0 when x > 0, y > 0.

7. Let a function f(z) be analytic in a domain Ω. Prove that f(z) must beconstant 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, orf(z) = c2/f(z) if |f(z) = c2 6= 0. Then use part (b).

8. Show that u(x, y) is harmonic in some domain and find a harmonic con-jugate v(x, y) when

(a) u(x, y) = 2x(1− y);

(b) u(x, y) = 2x− x3 + 3xy2;

(d) u(x, y) = y/(x2 + y2).

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) = c1 and v(x, y) = c2,where c1 and c2 are arbitrary real constants. Prove that these familiesare orthogonal. More precisely, show that if z0 = x0 + iy0 ∈ Ω is apoint common to two particular curves u(x, y) = c1 and v(x, y) = c2

and if f ′(z0) 6= 0, then the lines tangent to those curves at (x0, y0) areperpendicular.

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Suggestion: Note how it follows from the equations u(x, y) = c1 andv(x, y) = c2 that

∂u

∂x+

∂u

∂y

dy

dx= 0 and

∂v

∂x+

∂v

∂y

dy

dx= 0.

13. Show that when f(z) = z2, the level curves u(x, y) = c1 and v(x, y) = c2

of the component functions are hyperbolas asymptotic to the coordinateaxes, or to the lines y = x and y = −x. Note the orthogonality of the twofamilies, described in Exercise 12. Observe that the curves u(x, y) = 0and v(x, y) = 0 intersect at the origin but are not, however, orthogonal toeach 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 vwhen f(z) = 1/z, and note the orthogonality described in Exercise 12.

16. Sketch the families of level curves of the component functions u and vwhen

f(z) =z − 1z + 1

,

and note how the result in Exercise 12 is illustrated here.

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