Math 132 Section 1 Winter 2009
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Section: 1
Problem Set # 2
Problem (1)Show that for every complex number z,
Re(iz) = −Im(z).
Problem (2)If zis a complex number with Re(z)>1, show that 0 <Re(1/z)<1.
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Complex Analysis Problem Set 1, Winter 2009, Assignments of Mathematics
Problem set 1 for math 132, a college-level complex analysis course, from the winter semester of 2009. The problems cover various topics in complex analysis, including vector operations, vector projections, complex numbers, and complex exponentials.
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Math 132 Section 1 Winter 2009
Name:
Last First MI
Student ID
Section: 1
Problem Set # 2
Problem ( 1 ) Show that for every complex number z,
Re(iz) = −Im(z).
Problem ( 2 ) If z is a complex number with Re(z) > 1, show that 0 < Re(1/z) < 1.
Problem ( 3 ) What is to follow may or may not be of use but is anyway true. Let z 1 = a + ib and z 2 = c + id; a, b, c and d real. Consider the corresponding vectors which we will regard as 3–component vectors with the third component zero:
−→ W 1 = 〈a, b, 0 〉 −→ W 2 = 〈c, d, 0 〉
Show that both the real and imaginary parts of z 1 z 2 can be expressed in terms of the usual vector and scalar products of
W 1 and
W 2.
Problem ( 4 ) Let
A = 〈a, b〉 denote a two–component vector and ˆα = 〈cos α, sin α〉 a two–component unit vector. Recall the usual formula/definition of vector projection
Projαˆ(
A ) = (
A · αˆ)ˆα.
Let us identify
A with the complex number z = a + ib and, similarly, ˆα will be represented by eiα. Derive a formula for the corresponding complex number that represents Projαˆ(
A ) in terms of z, z and eiα.
Problem ( 8 ) Show that |z| ≤ |Re(z)| + |Im(z)| ≤
2 |z|.
Problem ( 9 ) Let a 0 , a 1 ,... an denote real constants and suppose that z is a solution to the equation
a 0 + a 1 z + ... + anzn^ = 0.
Show that ¯z is also a solution to this equation.
Problem ( 10 ) Write the complex number
e1+3iπ e−1+i^ π 2
in the form a + ib with a and b real.
Problem ( 11 ) Write the complex number ee i
in the form a + ib with a and b real.
Problem ( 12 ) Write the complex number
(1 + i)^6
in the polar form reiθ.
Problem ( 16 ) Let z be any complex number with modulus one except z = 1: |z| = 1, z 6 = 1. Show that
Re
1 − z
Problem ( 17 ) Let α denote an mth^ root of unity and β an mth^ root of unity (where, of course, m and n are integers). Show that the product is a kth^ root of unity for some integer k.
Problem ( 18 ) Find all solutions to the equation
z^4 =
z 5 + 3i
(You may express your solution in terms of some particular angle, ϕ, whose cosine you know.)
- Problem ( 19 ) Find all solutions to the equation z^5 = (z + 1)