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Mathematics

Complex Analysis

Branch Points and Branch Cuts

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f(z) = \log(z - a) + \log(z - b)

Branch points at $z = a$ and $z = b$ , branch cut commonly from $a$ to $b$ directly.

f(z) = (z - a)^b, (a \in \mathbb{C}, b \not\in \mathbb{Z})

Branch point at $z = a$ , branch cut commonly from $a$ to $\infty$ in a direction that avoids crossing other important points or cuts.

f(z) = \sqrt{1/z}

Branch point at $z = 0$ and infinity, branch cut commonly from $0$ to $\infty$ along the positive real axis.

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

Branch points at $z = 1$ and $z = -1$ , branch cut commonly from $-1$ to $1$ along the real axis.

f(z) = z^{1/3}

Branch point at $z = 0$ , branch cut commonly from $0$ to $-\infty$ along the negative real axis.

f(z) = \ln(z^2 - p^2), (p \in \mathbb{R}, p \neq 0)

Branch points at $z = p$ and $z = -p$ , branch cuts from each branch point to infinity, often chosen to be parallel to the real axis.

f(z) = \arcsin(z)

Branch points at $z = 1$ and $z = -1$ , suitable branch cuts from each point to infinity in such a way that ensures a principal branch can be defined.

f(z) = \tan^{-1}(z)

Branch points at $z = i$ and $z = -i$ , branch cut from $i$ to $-i$ usually taken along the imaginary axis.

f(z) = \cosh^{-1}(z)

Branch point at $z = 1$ , branch cut commonly from $1$ to $\infty$ along the real axis.

f(z) = \log(z^2 + 1)

Branch points at $z = i$ and $z = -i$ , suggested branch cut along the imaginary axis from $-i$ to $i$ or two cuts each extending from $i$ and $-i$ to infinity along the imaginary axis.

f(z) = \sqrt{z(z - 1)}

Branch points at $z = 0$ and $z = 1$ , branch cuts commonly from $0$ to $1$ directly or via a curve.

f(z) = \sqrt{z(z - a)^2}, (a \in \mathbb{C})

Branch points at $z = 0$ and $z = a$ , branch cut commonly from $0$ to $a$ , and from $a$ to $\infty$ or around a semicircle at infinity.

f(z) = \log(z)

Branch point at $z = 0$ , branch cut commonly from $0$ to $-\infty$ along the negative real axis.

f(z) = \sqrt[4]{z}

Branch point at $z = 0$ , and at infinity, branch cut commonly from $0$ to $-\infty$ along the negative real axis.

f(z) = \cos^{-1}(z)

Branch points at $z = 1$ and $z = -1$ , branch cuts usually from $-1$ to $1$ along the real axis.

f(z) = \sinh^{-1}(z)

No branch points, the function is single-valued.

f(z) = \frac{1}{\sqrt{z - a}}

Branch point at $z = a$ , branch cut commonly from $a$ to $\infty$ along a line or curve not crossing other points of interest.

f(z) = z^{a}, (a \not\in \mathbb{Z})

Branch point at $z = 0$ , branch cut commonly from $0$ to $-\infty$ along the negative real axis.

f(z) = \sqrt{z^2 + p^2}, (p \in \mathbb{R}^+)

Branch points at $z = ip$ and $z = -ip$ , branch cuts commonly from each branch point to infinity along lines parallel to the real axis.

f(z) = (z^2 + 1)^{1/4}

Branch points at $z = i$ and $z = -i$ , branch cuts from $i$ to $-i$ directly or along the imaginary axis to infinity.

f(z) = \sqrt{z(z - a)(z - b)}, (a \neq b)

Branch points at $z = 0$ , $z = a$ , and $z = b$ , branch cuts from $0$ to $a$ , and from $b$ to infinity, or others as per convention.

f(z) = (z^3 - 1)^{1/3}

Branch points at cube roots of unity ( $z = 1, z = -\frac{1}{2} + \frac{\sqrt{3}}{2}i, z = -\frac{1}{2} - \frac{\sqrt{3}}{2}i$ ), branch cuts from each root to infinity.

f(z) = \exp(\log(z))

Branch point at $z = 0$ , branch cut from $0$ to $-\infty$ along the negative real axis.

f(z) = \arctan(z)

Branch points at $z = i$ and $z = -i$ , branch cut from $i$ to $-i$ along the imaginary axis.

f(z) = (z^3 - 1)^{-1/3}

Branch points at the cube roots of unity ( $z = 1$ , $z = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$ , $z = -\frac{1}{2} - \frac{\sqrt{3}}{2}i$ ), branch cuts from each root to infinity, typically chosen to run parallel to the real or imaginary axes.

f(z) = \exp(\sqrt{z})

Branch point at $z = 0$ , branch cut commonly from $0$ to $-\infty$ along the negative real axis.

f(z) = (z - a)^{-2/3}

Branch point at $z = a$ , branch cut commonly from $a$ to $\infty$ omitting a line from the z-plane.

f(z) = (z^2 - 1)^{1/2}

Branch points at $z = 1$ and $z = -1$ , branch cuts commonly join these points to infinity directly or via a semicircle at infinity.

f(z) = z^{i}, (i \text{ is the imaginary unit})

Branch point at $z = 0$ , branch cut from $0$ to $-\infty$ along the negative real axis.

f(z) = \sqrt{z}

Branch point at $z = 0$ , branch cut commonly from $0$ to $-\infty$ along the negative real axis.

f(z) = \sqrt{(z - a)(z - b)}, (a \neq b)

Branch points at $z = a$ and $z = b$ , branch cut commonly connects $a$ and $b$ or extends from them to infinity.

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