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 $-2\le \mathrm{\Re }(k^2)\le 2$, $-2\le \mathrm{\Im }(k^2)\le 2$. 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 $-2\le \mathrm{\Re }(k^2)\le 2$, $-2\le \mathrm{\Im }(k^2)\le 2$. There is a branch cut where . Magnify 3D Help

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Figure 19.3.9: $\mathrm{\Re }(K\left(k\right))$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }(k^2)\le 2$, $-2\le \mathrm{\Im }(k^2)\le 2$. …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: $\mathrm{\Im }(K\left(k\right))$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }(k^2)\le 2$, $-2\le \mathrm{\Im }(k^2)\le 2$. …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: $\mathrm{\Im }(E\left(k\right))$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }(k^2)\le 2$, $-2\le \mathrm{\Im }(k^2)\le 2$. …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: 4.37 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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5: 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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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: 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 …

9: 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 $-\mathrm{i}\mathrm{\infty }$. …

10: Bibliography K

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Branch Cuts for Complex Elementary Functions or Much Ado About Nothing’s Sign Bit. In The State of the Art in Numerical Analysis (Birmingham, 1986), A. Iserles and M. J. D. Powell (Eds.), Inst. Math. Appl. Conf. Ser. New Ser., Vol. 9, pp. 165–211.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§4.23(iv), §4.37(iv).

Encodings:

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