Math 152: Worksheet 26

Complex Numbers

Learning Problems

These problems should be completed on your own. If you need hints on solving a problem, there are some provided with each problem. Click on the word "hint" to view it and again to hide it. They go in increasing order of helpfulness, with the last hint mostly giving away how to do the problem. Try to work from the earlier hints to the later ones, as this will give you the practice you need to succeed in this class.

Problem 1

Given complex numbers $z_1=3-4i$, and $z_2=2+i$, compute $2z_1-z_2$, $z_1\cdot z_2$ and $z_2-z_1$

• Hint 1

  For addition and subtraction, we need to add and subtract the real and imaginary parts separately.

• Hint 2

  When multiplying by a real number, we can just distribute that number to each of the real and imaginary parts.

• Hint 3

  For multiplying complex numbers, we distribute everything out, treating $i$ like a variable, and then using the fact that $i^2=-1$ to simplify the expression.

• Answer

  $2z_1-z_2=4-9i$, $z_1\cdot z_2=10-5i$ and $z_2-z_1=-1+5i$.

Problem 2

For $z_1=3-i$ and $z_2=2+3i$, calculate $\overline{z_1}$, $\overline{z_2}$, ${\displaystyle \frac{1}{z_1}}$ and ${\displaystyle \frac{z_2}{z_1}}$

• Hint 1

  To find the complex conjugate, we need to switch the sign of the imaginary part.

• Hint 2

  For any complex number ${\displaystyle \frac{1}{z}=\frac{\overline{z}}{|z{|}^2}}$.

• Hint 3

  Once you get $\frac{1}{z}$, the quotient of two complex numbers can be found by multiplying them.

• Hint 4

  For $z_1$, we get that ${\displaystyle \frac{1}{z_1}=\frac{3+i}{10}=\frac{3}{10}+\frac{1}{10}i}$

• Answer

  $\overline{z_1}=3+i$, $\overline{z_2}=2-3i$, ${\displaystyle \frac{1}{z_1}=\frac{3}{10}+\frac{1}{10}i}$ and ${\displaystyle \frac{z_2}{z_1}=\frac{3}{10}+\frac{11}{10}i}$

Problem 3

Convert the complex number $z=-2\sqrt{3}+2i$ into exponential form, and find the exponential form of $\overline{z}$.

• Hint 1

  We need to find $r$ and $\theta$ for the point $(-2\sqrt{3},2)$ in Cartesian coordinates.

• Hint 2

  We get that $r^2=(-2\sqrt{3}{)}^2+2^2=16$ and ${\displaystyle \theta ={\tan }^{-1}\left(\frac{2}{-2\sqrt{3}}\right)={\tan }^{-1}(-\frac{1}{\sqrt{3}})}$

• Hint 3

  So $r=4$ and ${\displaystyle \theta =-\frac{\pi }{6}}$, but this $\theta$ is in the wrong quadrant. So we actually want ${\displaystyle \frac{5\pi }{6}}$ for $\theta$.

• Hint 4

  To get $\overline{z}$, we need to reflect over the $x$ axis. This results in taking the negative of the $\theta$ angle. We can also just start with $\overline{z}=-2\sqrt{3}-2i$ and go from there.

• Answer

  ${\displaystyle z=4e^{i\frac{5\pi }{6}}}$, ${\displaystyle \overline{z}=4e^{-i\frac{5\pi }{6}}=4e^{i\frac{7\pi }{6}}}$

Problem 4

Find the partial fraction decomposition of ${\displaystyle \frac{6x^4-13x^3+37x^2-37x-1}{(x-2)(x^2+1)(x^2+9)}}$.

• Hint 1

  The decomposition here looks like $\frac{A}{x-2}+\frac{Bx+C}{x^2+1}+\frac{Dx+E}{x^2+9}$

• Hint 2

  To set up value substitution, we end up with ${\displaystyle 6x^4-13x^3+37x^2-37x-1=A(x^2+1)(x^2+9)+(Bx+C)(x-2)(x^2+9)+(Dx+E)(x-2)(x^2+1)}$.

• Hint 3

  To set this up, we need to plug in $x=2$, $x=i$, and $x=3i$.

• Hint 4

  Each of the complex numbers gives rise to a system of two equations to solve for two coefficients, with these equations coming from the real and imaginary parts of the complex numbers.

• Answer

  ${\displaystyle \frac{1}{(x-2)}+\frac{2x+1}{x^2+1}+\frac{3x-4}{x^2+9}}$

Problem 5

Find the four complex numbers where the function ${\displaystyle f(x)=\frac{1}{(x^2+1)(x^2+2x+5)}}$ does not exist, and use this to determine an upper bound on the radius of convergence of the power series expansion of $f(x)$ centered at $x=-3$.

• Hint 1

  The denominator of this function is $(x^2+1)(x^2+2x+5)$, so we can find where this part is zero.

• Hint 2

  The roots we get here are $z_1=i$, $z_2=-i$, and $z_3=-1+2i$, $z_4=-1-2i$.

• Hint 3

  To find the radius of convergence, or an upper bound on it, we need to figure out how far each of these numbers are from $-3$, because the radius can not extend past any of these points.

• Hint 4

  The distances are $\sqrt{10}$, $\sqrt{10}$, $\sqrt{8}$ and $\sqrt{8}$.

• Answer

  $f(x)$ does not exist at $z_1=i$, $z_2=-i$, and $z_3=-1+2i$, $z_4=-1-2i$. An upper bound on the radius of convergence is $\sqrt{8}$.

Submission Problems

Problem 1

Use complex numbers to help compute ${\displaystyle \int \frac{x^2-5x+9}{(x+1)(x^2+4)}dx}$

Problem 2

Use complex numbers to find an upper bound on the radius of convergence of the power series expansion of $\frac{\sin x}{x^2+9}$ centered at $x=1$. Do not compute the power series expansion.