Problems for Lecture 5
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In Problems 1 and 2 write each of the given numbers in the form a + bi.
1(c). ee
i
2(b). 2e3+i/6
In Problems 3 and 4 write each of the given numbers in the polar form rei .
3(b). 8(1 +
3i)
3(c). (1 + i)6
4(b).
2+2i
3+i
5.
Problems for Lecture 14
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1. Show that ez = (1 + i)/ 2 if and only if z = (/4 + 2k)i, k = 0, 1, 2, . . .
5. Write each of the following numbers in the form a + bi.
exp(1+i3)
(a) exp(2 + i/4)
(b) exp(1+i/2)
9(a). Find dw/dz for w = exp(z 2 ).
17(b)
Problems for Lecture 15
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1. Evaluate each of the following.
(b) log(1 i)
(c) Log(i)
3. Show that if z1 = i and z2 = i 1, then
Logz1 z2 6= Logz1 + Logz2 .
5(b). Solve the equation Log(z 2 1) =
i
.
2
6. Find the error in the following proof that z
Problems for Lecture 26
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2. Let f and g be analytic inside and on the simple loop . Prove that if f (z) = g(z) for all
z on , then f (z) = g(z) for all z inside .
3(f). Let C be the circle |z| = 2 traversed once in the positive sense. Compute
R
s
Problems for Lecture 16
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5(d). Write each of the following numbers in the form a + bi.
(d) cos(1 i)
(f) cosh(i/2)
6. Establish the trigonometric identities
sin2 z + cos2 z = 1
(8)
sin(z1 z2 ) = sin z1 cos z2 sin z2 cos z1 .
(9)
and
7. Show that t
Problems for Lecture 19
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1. For each of the following smooth curves give an admissible parametrization that is consistent with the indicated direction.
(a) the line segment from z = 1 + i to z = 2 3i
(b) the circle |z 2i| = 4 traversed once in th
Problems for Lecture 30
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8. If f is analytic in the annulus 1 |z| 2 and |f (z)| 3 on |z| = 1 and |f (z)| 12 on
|z| = 2, prove that |f (z)| 3|z|2 for 1 |z| 2. [HINT: Consider f (z)/3z 2 .]
11. Suppose that f is analytic inside and on the simple cl
Problems for Lecture 28
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1. Let f (z) = 1/(1 z)2 , and let 0 < R < 1. Verify that max |f (z)| = 1/(1 R)2 , and also
|z|=R
show f
(n+1)
(0) = (n + 1)!, so that by the Cauchy estimates
(n + 1)|
n!
.
R)2
Rn (1
3. Let f be analytic and bounded by M
Problems for Lecture 8
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1. Write each of the following functions in the form w = u(x, y) + iv(x, y).
(c) h(z) =
z+i
z 2 +1
(d) q(z) =
2z 2 +3
|z1|
(e) F (z) = e3z
2. Find the domain of definition of each of the functions in Prob. 1.
3. Describe th
Problems for Lecture 10
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4(c). Use Definition 4, show that the function f (z) = |z| is nowhere differentiable.
[Remark: Definition 4 is the definition for the derivative of a function at a point.]
7. Use rules (5)(9) to find the derivatives of the
Problems for Lecture 1
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In Problems 513, write the number in the form a + bi.
5(b). (8 + i) (5 + i)
6(a). (1 + i)2
11. i3 (i + 1)2
14. Show that Re(iz) = Imz for every complex number z.
19. Write the complex equation z 3 + 5z 2 = z + 3i as two real
Problems for Lecture 2
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9. Write the number
2+3i
1+2i
8+i
6i
in the form a + bi.
17. Use the result of Problem 15 to evaluate
3i11 + 6i3 +
8
+ i1 .
20
i
[15. Let k be an integer. Show that
i4k = 1,
i4k+1 = i,
i4k+2 = 1,
i4k+3 = i.
]
21. The complex
Problems for Lecture 3
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3. Which of the points i, 2 i, and 3 is farthest from the origin?
4. Let z = 3 2i. Plot the points z, z, z,
z , and 1/z in the complex plane. Do the same
for z = 2 + 3i and z = 2i.
5. Show that the points 1, 1/2 + i 3/2, a
Problems for Lecture 4
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7. Find the argument of each of the following complex numbers and write each in polar form.
3i
(g) 1+
(e) (1 i)( 3 + i)
2+2i
12. Find the following.
(a) Arg(6 6i)
(b) Arg()
13. Decide which of the following statements are a
Mathematics 304 Assignment 2
Solutions
Due: Friday, November 22, 2013 at the beginning of class
1. Let be any contour from 1 to i. Evaluate
Z
z cosh(z)dz.
Solution 1: Note that z cosh(z) is continuous everywhere and
d
(z sinh(z) cosh(z) = z cosh(z)
dz
for
Problems for Lecture 6
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5. Find all the values of the following.
(a) (16)1/4
(e) (i 1)1/2
2i 1/6
)
(f) ( 1+i
7(b). Solve the equation z 2 (3 2i)z + 1 3i = 0.
9. Solve the equation z 3 3z 2 + 6z 4 = 0.
11. Solve the equation (z + 1)5 = z 5 .
15. Us
Problems for Lecture 9
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8. Use Definition 2 to prove that limz1+i (6z 4) = 2 + 6i.
[ Remark: Definition 2 uses the language to define the limit of a function at a point.]
11. Find each of the following limits.
2
(b) lim z iz+3
z2
(z0 +z)2 z02
z
z0
Problems for Lecture 13
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1(b). Verify directly that the real and imaginary parts of the analytic function g(z) =
1
z
satisfy
Laplaces equation.
3. Verify that each given function u is harmonic (in the region where it is defined) and then
find a ha