# branch cuts

(0.000 seconds)

## 1—10 of 28 matching pages

##### 1: 4.2 Definitions Figure 4.2.1: z -plane: Branch cut for ln ⁡ z and z α . Magnify Most texts extend the definition of the principal value to include the branch cut
##### 2: 4.15 Graphics Figure 4.15.9: arcsin ⁡ ( x + i ⁢ y ) (principal value). There are branch cuts along the real axis from − ∞ to − 1 and 1 to ∞ . Magnify 3D Help Figure 4.15.11: arctan ⁡ ( x + i ⁢ y ) (principal value). There are branch cuts along the imaginary axis from − i ⁢ ∞ to − i and i to i ⁢ ∞ . Magnify 3D Help Figure 4.15.13: arccsc ⁡ ( x + i ⁢ y ) (principal value). There is a branch cut along the real axis from − 1 to 1 . Magnify 3D Help
##### 3: 19.3 Graphics Figure 19.3.7: K ⁡ ( k ) as a function of complex k 2 for − 2 ≤ ℜ ⁡ ( k 2 ) ≤ 2 , − 2 ≤ ℑ ⁡ ( k 2 ) ≤ 2 . There is a branch cut where 1 < k 2 < ∞ . Magnify 3D Help Figure 19.3.8: E ⁡ ( k ) as a function of complex k 2 for − 2 ≤ ℜ ⁡ ( k 2 ) ≤ 2 , − 2 ≤ ℑ ⁡ ( k 2 ) ≤ 2 . There is a branch cut where 1 < k 2 < ∞ . Magnify 3D Help Figure 19.3.9: ℜ ⁡ ( K ⁡ ( k ) ) as a function of complex k 2 for − 2 ≤ ℜ ⁡ ( k 2 ) ≤ 2 , − 2 ≤ ℑ ⁡ ( k 2 ) ≤ 2 . …On the branch cut ( k 2 ≥ 1 ) it is infinite at k 2 = 1 , and has the value K ⁡ ( 1 / k ) / k when k 2 > 1 . Magnify 3D Help Figure 19.3.10: ℑ ⁡ ( K ⁡ ( k ) ) as a function of complex k 2 for − 2 ≤ ℜ ⁡ ( k 2 ) ≤ 2 , − 2 ≤ ℑ ⁡ ( k 2 ) ≤ 2 . …On the upper edge of the branch cut ( k 2 ≥ 1 ) it has the value K ⁡ ( k ′ ) if k 2 > 1 , and 1 4 ⁢ π if k 2 = 1 . Magnify 3D Help Figure 19.3.12: ℑ ⁡ ( E ⁡ ( k ) ) as a function of complex k 2 for − 2 ≤ ℜ ⁡ ( k 2 ) ≤ 2 , − 2 ≤ ℑ ⁡ ( k 2 ) ≤ 2 . …On the upper edge of the branch cut ( k 2 > 1 ) it has the (negative) value K ⁡ ( k ′ ) − E ⁡ ( k ′ ) , with limit 0 as k 2 → 1 + . Magnify 3D Help
##### 4: 4.3 Graphics Figure 4.3.3: ln ⁡ ( x + i ⁢ y ) (principal value). There is a branch cut along the negative real axis. Magnify 3D Help
##### 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. …
##### 6: 8.19 Generalized Exponential Integral Figure 8.19.2: E 1 2 ⁡ ( x + i ⁢ y ) , − 4 ≤ x ≤ 4 , − 4 ≤ y ≤ 4 . …There is a branch cut along the negative real axis. Magnify 3D Help Figure 8.19.3: E 1 ⁡ ( x + i ⁢ y ) , − 4 ≤ x ≤ 4 , − 4 ≤ y ≤ 4 . …There is a branch cut along the negative real axis. Magnify 3D Help Figure 8.19.4: E 3 2 ⁡ ( x + i ⁢ y ) , − 3 ≤ x ≤ 3 , − 3 ≤ y ≤ 3 . …There is a branch cut along the negative real axis. Magnify 3D Help Figure 8.19.5: E 2 ⁡ ( x + i ⁢ y ) , − 3 ≤ x ≤ 3 , − 3 ≤ y ≤ 3 . …There is a branch cut along the negative real axis. Magnify 3D Help
##### 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. …
##### 8: 4.13 Lambert $W$-Function Figure 4.13.2: The W ⁡ ( z ) function on the first 5 Riemann sheets. … Magnify where $t\geq 0$ for $W_{0}$, $t\leq 0$ 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 $+\infty$ on the real $z$-axis, that is, the branch in the sector $|\operatorname{ph}\left(1-z\right)|\leq\pi$, is the principal branch (or principal value) of $F\left(a,b;c;z\right)$. … The difference between the principal branches on the two sides of the branch cut4.2(i)) is given by …
##### 10: 3.10 Continued Fractions
For example, by converting the Maclaurin expansion of $\operatorname{arctan}z$ (4.24.3), we obtain a continued fraction with the same region of convergence ($\left|z\right|\leq 1$, $z\neq\pm\mathrm{i}$), whereas the continued fraction (4.25.4) converges for all $z\in\mathbb{C}$ except on the branch cuts from $i$ to $i\infty$ and $-i$ to $-i\infty$. …