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Complex Analysis

Complex Zeros and Singularities

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f(z) = \sinh(z)

Zeros: z = n\pi i, n \in \mathbb{Z}; Singularities: None

f(z) = \frac{z^2 + 1}{z^3 - 1}

Zeros: z = i, z = -i; Singularities: z = 1, z = -\frac{1}{2} + \frac{\sqrt{3}}{2}i, z = -\frac{1}{2} - \frac{\sqrt{3}}{2}i (simple poles)

f(z) = \frac{1}{\sinh(z)}

Zeros: None; Singularities: z = n\pi i, n \in \mathbb{Z} (simple poles)

f(z) = e^{z^2}

Zeros: None; Singularities: None

f(z) = \frac{z}{(z - 1)^2}

Zeros: z = 0; Singularities: z = 1 (pole of order 2)

f(z) = \frac{1}{z^3 + 1}

Zeros: None; Singularities: z = -1, z = \frac{1}{2} - \frac{\sqrt{3}}{2}i, z = \frac{1}{2} + \frac{\sqrt{3}}{2}i (poles of first order)

f(z) = \cosh(z)

Zeros: None; Singularities: None

f(z) = z^2e^\frac{1}{z}

Zeros: z = 0 (of order 2); Singularities: z = 0 is also an essential singularity

f(z) = \log(z)

Zeros: None; Singularities: z = 0 (branch point), along the negative real axis (branch cut)

f(z) = \tan(z)

Zeros: z = n\pi, n \in \mathbb{Z}; Singularities: z = \frac{\pi}{2} + n\pi, n \in \mathbb{Z} (poles of first order)

f(z) = z^3

Zeros: z = 0 (of order 3); Singularities: None

f(z) = \frac{z - 2}{z^2 + z + 1}

Zeros: z = 2; Singularities: z = \frac{-1 \pm i\sqrt{3}}{2} (complex conjugate pair of poles)

f(z) = \frac{z^2 + 2z + 2}{z^2 - 4}

Zeros: z = -1 \pm i; Singularities: z = 2, z = -2 (simple poles)

f(z) = \frac{1}{\cos(z)}

Zeros: None; Singularities: z = \frac{\pi}{2} + n\pi, n \in \mathbb{Z} (poles of first order)

f(z) = \frac{z}{e^z - 1}

Zeros: z = 0; Singularities: z = 2n\pi i, n \in \mathbb{Z} \setminus \{0\} (simple poles)

f(z) = e^{\frac{1}{z^2}}

Zeros: None; Singularities: z = 0 (essential singularity)

f(z) = e^z - 1

Zeros: z = 2n\pi i, n \in \mathbb{Z}; Singularities: None

f(z) = z^3 - 1

Zeros: z = 1, z = -\frac{1}{2} + \frac{\sqrt{3}}{2}i, z = -\frac{1}{2} - \frac{\sqrt{3}}{2}i; Singularities: None

f(z) = \sin(\frac{1}{z})

Zeros: z = \frac{1}{n\pi}, n \in \mathbb{Z} \setminus \{0\}; Singularities: z = 0 (essential singularity)

f(z) = \frac{1}{z - i}

Zeros: None; Singularities: z = i (simple pole)

f(z) = \frac{z}{\cosh(z)}

Zeros: z = (2n + 1)\frac{\pi i}{2}, n \in \mathbb{Z}; Singularities: None

f(z) = \cos(\pi z)

Zeros: z = n + \frac{1}{2}, n \in \mathbb{Z}; Singularities: None

f(z) = \frac{1}{z^4 + 1}

Zeros: None; Singularities: z = e^{\frac{i\pi}{4}}, e^{\frac{3i\pi}{4}}, e^{\frac{5i\pi}{4}}, e^{\frac{7i\pi}{4}} (poles of first order)

f(z) = \frac{z - i}{z^2 + 1}

Zeros: z = i; Singularities: z = i, z = -i (simple pole at z = i is canceled)

f(z) = \frac{sin(z)}{z^2 + 2z + 1}

Zeros: z = n\pi, n \in \mathbb{Z} \setminus \{0\}; Singularities: z = -1

f(z) = \frac{z}{\sin(\pi z)}

Zeros: z = n, n \in \mathbb{Z}; Singularities: z = n + \frac{1}{2}, n \in \mathbb{Z} (simple poles)

f(z) = \frac{z^2}{\sin(z^2)}

Zeros: z = 0; Singularities: z = \pm\sqrt{n\pi}, n \in \mathbb{Z} \setminus \{0\} (simple poles)

f(z) = e^{\sin(z)}

Zeros: None; Singularities: None

f(z) = \frac{1}{1 + z^2}

Zeros: None; Singularities: z = i, z = -i

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