About the Project

other branches

AdvancedHelp

(0.002 seconds)

1—10 of 95 matching pages

1: 4.13 Lambert W -Function
We call the increasing solution for which W ( z ) W ( e 1 ) = 1 the principal branch and denote it by W 0 ( z ) . …
See accompanying text
Figure 4.13.1: Branches W 0 ( x ) , W ± 1 ( x 0 i ) of the Lambert W -function. Magnify
Other solutions of (4.13.1) are other branches of W ( z ) . …The other branches W k ( z ) are single-valued analytic functions on ( , 0 ] , have a logarithmic branch point at z = 0 , and, in the case k = ± 1 , have a square root branch point at z = e 1 0 i respectively. …
2: 4.24 Inverse Trigonometric Functions: Further Properties
3: 1.10 Functions of a Complex Variable
If D = ( , 0 ] and z = r e i θ , then one branch is r e i θ / 2 , the other branch is r e i θ / 2 , with π < θ < π in both cases. Similarly if D = [ 0 , ) , then one branch is r e i θ / 2 , the other branch is r e i θ / 2 , with 0 < θ < 2 π in both cases. … (b) By specifying the value of F ( z ) at a point z 0 (not a branch point), and requiring F ( z ) to be continuous on any path that begins at z 0 and does not pass through any branch points or other singularities of F ( z ) . If the path circles a branch point at z = a , k times in the positive sense, and returns to z 0 without encircling any other branch point, then its value is denoted conventionally as F ( ( z 0 a ) e 2 k π i + a ) . …
4: 15.2 Definitions and Analytical Properties
again with analytic continuation for other values of z , and with the principal branch defined in a similar way. … The same is true of other branches, provided that z = 0 , 1 , and are excluded. …
5: 4.37 Inverse Hyperbolic Functions
Arcsinh z and Arccsch z have branch points at z = ± i ; the other four functions have branch points at z = ± 1 . …
6: 10.21 Zeros
For further information, including uniform asymptotic expansions, extensions to other branches of the functions and their derivatives, and extensions to half-integer values of the order, see Olver (1954). …
7: 31.11 Expansions in Series of Hypergeometric Functions
The expansion (31.11.1) for a Heun function that is associated with any branch of (31.11.2)—other than a multiple of the right-hand side of (31.11.12)—is convergent inside the ellipse . …
8: 14.24 Analytic Continuation
For fixed z , other than ± 1 or , each branch of P ν μ ( z ) and 𝑸 ν μ ( z ) is an entire function of each parameter ν and μ . …
9: 4.23 Inverse Trigonometric Functions
Arctan z and Arccot z have branch points at z = ± i ; the other four functions have branch points at z = ± 1 . …
10: 28.7 Analytic Continuation of Eigenvalues
The branch points are called the exceptional values, and the other points normal values. …