branch cuts

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1: 4.2 Definitions

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Figure 4.2.1: $z$-plane: Branch cut for $\ln z$ and $z^{\alpha }$. Magnify

… ►Most texts extend the definition of the principal value to include the branch cut …

2: 4.15 Graphics

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Figure 4.15.9: $\arcsin \left(x+\mathrm{i}y\right)$ (principal value). There are branch cuts along the real axis from $-\mathrm{\infty }$ to $-1$ and $1$ to $\mathrm{\infty }$. Magnify 3D Help

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Figure 4.15.11: $\arctan \left(x+\mathrm{i}y\right)$ (principal value). There are branch cuts along the imaginary axis from $-\mathrm{i}\mathrm{\infty }$ to $-\mathrm{i}$ and $\mathrm{i}$ to $\mathrm{i}\mathrm{\infty }$. Magnify 3D Help

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Figure 4.15.13: $\mathrm{arccsc}\left(x+\mathrm{i}y\right)$ (principal value). There is a branch cut along the real axis from $-1$ to $1$. Magnify 3D Help

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3: 19.3 Graphics

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Figure 19.3.7: $K\left(k\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. There is a branch cut where . Magnify 3D Help

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Figure 19.3.8: $E\left(k\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. There is a branch cut where . Magnify 3D Help

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Figure 19.3.9: $\mathrm{\Re }\left(K\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the branch cut ($k^2\ge 1$) it is infinite at $k^2=1$, and has the value $K(1/k)/k$ when $k^2>1$. Magnify 3D Help

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Figure 19.3.10: $\mathrm{\Im }\left(K\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the upper edge of the branch cut ($k^2\ge 1$) it has the value $K\left(k^{\prime }\right)$ if $k^2>1$, and $\frac{1}{4}\pi$ if $k^2=1$. Magnify 3D Help

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Figure 19.3.12: $\mathrm{\Im }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the upper edge of the branch cut ($k^2>1$) it has the (negative) value $K\left(k^{\prime }\right)-E\left(k^{\prime }\right)$, with limit 0 as $k^2\to 1+$. Magnify 3D Help

4: 4.3 Graphics

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Figure 4.3.3: $\ln \left(x+\mathrm{i}y\right)$ (principal value). There is a branch cut along the negative real axis. Magnify 3D Help

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5: 4.37 Inverse Hyperbolic Functions

… ►The principal values (or principal branches) of the inverse $\sinh$, $\cosh$, and $\tanh$ are obtained by introducing cuts in the $z$-plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts. … ►

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Figure 4.37.1: $z$-plane. Branch cuts for the inverse hyperbolic functions.

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6: 8.19 Generalized Exponential Integral

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Figure 8.19.2: $E_{\frac{1}{2}}\left(x+\mathrm{i}y\right)$, $-4\le x\le 4$, $-4\le y\le 4$. …There is a branch cut along the negative real axis. Magnify 3D Help

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Figure 8.19.3: $E_1\left(x+\mathrm{i}y\right)$, $-4\le x\le 4$, $-4\le y\le 4$. …There is a branch cut along the negative real axis. Magnify 3D Help

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Figure 8.19.4: $E_{\frac{3}{2}}\left(x+\mathrm{i}y\right)$, $-3\le x\le 3$, $-3\le y\le 3$. …There is a branch cut along the negative real axis. Magnify 3D Help

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Figure 8.19.5: $E_2\left(x+\mathrm{i}y\right)$, $-3\le x\le 3$, $-3\le y\le 3$. …There is a branch cut along the negative real axis. Magnify 3D Help

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7: 4.23 Inverse Trigonometric Functions

… ►The principal values (or principal branches) of the inverse sine, cosine, and tangent are obtained by introducing cuts in the $z$-plane as indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the integration paths in (4.23.1)–(4.23.3) not to cross these cuts. … ►

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Figure 4.23.1: $z$-plane. Branch cuts for the inverse trigonometric functions.

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8: 4.13 Lambert $W$-Function

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Figure 4.13.2: The $W\left(z\right)$ function on the first 5 Riemann sheets. … Magnify

… ►where $t\ge 0$ for $W_0$, $t\le 0$ for $W_{\pm 1}$ on the relevant branch cuts, …

9: 15.2 Definitions and Analytical Properties

… ►The branch obtained by introducing a cut from $1$ to $+\mathrm{\infty }$ on the real $z$-axis, that is, the branch in the sector $|\mathrm{ph}\left(1-z\right)|\le \pi$, is the principal branch (or principal value) of $F(a,b;c;z)$. … ►The difference between the principal branches on the two sides of the branch cut (§4.2(i)) is given by …

10: 3.10 Continued Fractions

… ►For example, by converting the Maclaurin expansion of $\arctan z$ (4.24.3), we obtain a continued fraction with the same region of convergence ($\left|z\right|\le 1$, $z\ne \pm \mathrm{i}$), whereas the continued fraction (4.25.4) converges for all $z\in ℂ$ except on the branch cuts from $\mathrm{i}$ to $\mathrm{i}\mathrm{\infty }$ and $-\mathrm{i}$ to $-\mathrm{i}\mathrm{\infty }$. …