Mathematics for Engineering I

Brandenburg Technical University (BTU) Cottbus

Chair of Mathematics for Enineering Prof. Dr. R. Reemtsen, Dr. F. Kemm

Mathematics for Engineering I

Problem Sheet No. 5, November 19/20, 2007 www.math.tu-cottbus.de/˜kemm/lehre/erm

Class Problems

1. Write the following numbers in Euler form:

z1 =−1

3(1−i), z2 =−10, z3 = (1 +i)⁹ .

2. Given the complex numbers z₁ = ¹₇ + 4i, z₂ = −3− ¹₆i, z₃ = −1 + 2i, write the following numbers in the forma+ib with a, b∈R:

w₁ =z₁+ 2z₂−3z₃ , w₂ =z₁z₂²z₃ , w₃ = z₁ z3

.

3. Solve the following equations in the complex plane:

z² =i , (1)

z⁴ =−16, (2)

z² =−5 + 12i , (3)

z²+ 4z+ 4 + 2i= 0. (4)

4. Sketch the following sets in the complex plane:

A:={z∈C| 1

2 <|z+ 2 +i|<2}, B :={z∈C| |z−1|+|z+ 1|= 6},

C :={z∈C| |z−a|=|z−b|}, a, b∈C given, a6=b

D:=

z ∈C|Re{z}= Im{z} , E :=

z ∈C| |Re{z}|+|Im{z}| ≤1 .

5. Given z =−1 + 2i, mark the following numbers in the complex plane:

iz , iz , iz , z ,

−z , −z , z² , 1

z .

Homework

1. Compute the complex square roots of z₁ =−7 andz₂ =−i−1.

Compute also the cubic roots ofz3 =i and z4 =−27.

2. Given z = 3 +i/2, mark the following numbers in the complex plane:

iz , iz , iz , z ,

−z , −z , z² , 1

z .

3. Simplify as much as you can:

(2−3i)·(−1 + 5i), 5

1−2i , 5 + 12i

3 + 2i , (1 +i)⁸ , (1−i)²(1 +i)³ , i²⁰⁰⁷ . 4. Compute the absolute values of the following complex numbers:

z₁ =−5 + 4i , z₂ =√

3 cos(10π) +isin(10π)

, z₃ = 1 +ai

1−ai , a ∈R. 5. Write the following numbers in Euler form:

z₁ =−√

3(−1 +i), z₂ =−11i , z₃ = (1−i)²4. Hint: First make a sketch of the numbers in the complex plane.

6. Write the following numbers in the form a+ib with a, b∈R: 2e^π3i , √

12e^−π3i . Mark the numbers in the complex plane.

7. Simplify as much as you can:

r a³

q a²√⁴

a³ , ⁴

v u u t

9a⁶ b²c

!n

· v u u t

27b⁵ a⁵√

c

!n

.