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Solving Complex Equations
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z^3 = 8
z = 2, z = -1 + i\sqrt{3}, z = -1 - i\sqrt{3}
z^2 = -4
z = 2i, z = -2i
z + 1/z = 2
z = 1
z^4 = 16
z = 2, z = -2, z = 2i, z = -2i
e^z = 1
z = 2k\pi i, k \in \mathbb{Z}
z^2 + 4 = 0
z = 2i, z = -2i
z^2 + z + 1 = 0
z = \frac{-1 + i\sqrt{3}}{2}, z = \frac{-1 - i\sqrt{3}}{2}
e^z = i
z = \frac{\pi}{2} + 2k\pi i, k \in \mathbb{Z}
z^2 - 6z + 9 = 0
z = 3
iz + 1 = 0
z = -\frac{1}{i} = i
z^2 = -1
z = i, z = -i
z^4 = -16
z = \sqrt[4]{16}e^{\frac{\pi i}{4}}, z = \sqrt[4]{16}e^{\frac{3\pi i}{4}}, z = \sqrt[4]{16}e^{\frac{5\pi i}{4}}, z = \sqrt[4]{16}e^{\frac{7\pi i}{4}}
z^2 + 1 = 0
z = i, z = -i
z^2 = 1 - i
z = \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}i, z = -\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i
iz^2 + z - i = 0
z = -i, z = 1
z^5 = 32i
z = 2e^{\frac{\pi i}{10}}, z = 2e^{\frac{3\pi i}{10}}, z = 2e^{\frac{5\pi i}{10}}, z = 2e^{\frac{7\pi i}{10}}, z = 2e^{\frac{9\pi i}{10}}
z^4 + 4 = 0
z = 1 + i, z = -1 - i, z = 1 - i, z = -1 + i
e^{2z} = 4
z = \log{2} + k\pi i, k \in \mathbb{Z}
e^{3z} = -1
z = \frac{1}{3}(\pi + 2k\pi i), k \in \mathbb{Z}
z^2 = 8 \times (1 + i)
z = 2 + 2i, z = -2 - 2i
z^3 = i
z = e^{\frac{\pi i}{6}}, z = e^{\frac{5\pi i}{6}}, z = e^{\frac{3\pi i}{2}}
z^3 = -27
z = -3, z = 3e^{\frac{\pi i}{3}}, z = 3e^{\frac{5\pi i}{3}}
z^2 - 2z + 2 = 0
z = 1 + i, z = 1 - i
e^{2z} + 1 = 0
z = i\frac{\pi}{2} + k\pi i, k \in \mathbb{Z}
e^{iz} = 1
z = 2k\pi, k \in \mathbb{Z}
z^8 = 256
z = 2, z = 2e^{\frac{\pi i}{4}}, z = 2e^{\frac{2\pi i}{4}}, z = 2e^{\frac{3\pi i}{4}}, z = -2, z = 2e^{\frac{5\pi i}{4}}, z = 2e^{\frac{6\pi i}{4}}, z = 2e^{\frac{7\pi i}{4}}
z^6 = 64
z = 2, z = 2e^{\frac{\pi i}{3}}, z = 2e^{\frac{2\pi i}{3}}, z = -2, z = 2e^{\frac{4\pi i}{3}}, z = 2e^{\frac{5\pi i}{3}}
z + \bar{z} = 4
z = 2 + bi, b \in \mathbb{R}
e^z = -i
z = -\frac{\pi}{2} + 2k\pi i, k \in \mathbb{Z}
z^6 + 1 = 0
z = e^{\frac{\pi i}{6}}, z = e^{\frac{3\pi i}{6}}, z = e^{\frac{5\pi i}{6}}, z = e^{\frac{7\pi i}{6}}, z = e^{\frac{9\pi i}{6}}, z = e^{\frac{11\pi i}{6}}
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