Complex Numbers

Section 17.1: 1, 4, 5, 10, 14, 16, 17, 19, 27, 30, 33, 36, 38, 42, 44, 45, 47, 48

• Polar form: $z = r(\cos \theta + i \sin \theta)$, where $r$ is the magnitude of the complex number

  

• Arg(z) is the angle between the complex number and the real axis

  

• Multiplication: $|z_1z_2| = |z_1||z_2|$

• Division: $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|} \angle \theta _2 - \theta _1$

  • Easier to use

Powers and Roots

Section 17.2: 1, 3, 9, 11, 14, 15, 18, 20, 21, 26, 27, 29, 30, 34, 36, 37, 38, 39, 40.

• Integer Powers of $z$: $z^n = r^n(\cos(n \theta + i \sin n \theta)$
• De Moivre’s Formula: $(\cos \theta + i \sin \theta)^n = \cos n \theta + i \sin n \theta$
  • If $z = re^{i \theta}$, then $z^n = r^n e^{in \theta}$
• Roots of a Complex Number: Complex numbers that when raised to the $n$th power, they equal the original number, such that $w^n = z$
  • $\sqrt[n]{z} = \sqrt[n]{r} (\cos \frac{\theta + 2 \pi k}{n} + i \sin \frac{\theta + 2 \pi k}{n})$

Exponential and Logarithmic Functions

Section 17.6: 1, 6, 11, 12, 15, 16, 17, 18, 20, 24, 28, 32, 33, 36, 38, 40, 41, 44, 45, 47ac, 48.

Review (pg. 862): 3, 9, 10, 15, 18.

• $e^z = e^{x+iy} = e^x(\cos y + i \sin y)$

  • Multiplication: $e^{z_1z_2} = e^{z_1}e^{z_2}$

  • Division: $\frac{e^{z_1}}{e^{z_2}} = e^{z_1-z_2}$

  • Periodicity: $f(z+2 \pi i) = f(z)$

  

  • Polar form: $z = re^{i \theta}$

• Logarithm: $\ln z = \log _e |z| + i(\theta + 2 \pi n), n = 0, \pm 1,2,3...$

  • Principal Value: $Ln z = \log _e |z| + i Arg z$

  • Multiplication: $\ln(z_1z_2) = \ln z_1 + \ln z_2$

    • Does not apply to $Ln$

  • Division: $\ln \frac{z_1}{z_2} =\ln z_1 - \ln z_2$

  • Analyticity: $\frac{d}{dz}Ln z = \frac{1}{z}$

    

    Arg(z) is only discontinuous at parts of the negative real axis.

  • Complex Powers: $z^{\alpha} = e^{\alpha \ln z}, z≠ 0$