About the Project

value at infinity

AdvancedHelp

(0.007 seconds)

11—20 of 132 matching pages

11: 4.31 Special Values and Limits
§4.31 Special Values and Limits
Table 4.31.1: Hyperbolic functions: values at multiples of 1 2 π i .
z 0 1 2 π i π i 3 2 π i
tanh z 0 i 0 i 1
sech z 1 1 0
coth z 0 0 1
12: 3.6 Linear Difference Equations
If, as n , the wanted solution w n grows (decays) in magnitude at least as fast as any solution of the corresponding homogeneous equation, then forward (backward) recursion is stable. … The least value of N that satisfies (3.6.9) is found to be 16. … For a difference equation of order k ( 3 ), …Typically k conditions are prescribed at the beginning of the range, and conditions at the end. …
13: 16.2 Definition and Analytic Properties
The branch obtained by introducing a cut from 1 to + on the real axis, that is, the branch in the sector | ph ( 1 z ) | π , is the principal branch (or principal value) of F q q + 1 ( 𝐚 ; 𝐛 ; z ) ; compare §4.2(i). Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at z = 0 , 1 , and . Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. … In general the series (16.2.1) diverges for all nonzero values of z . … (However, except where indicated otherwise in the DLMF we assume that when p > q + 1 at least one of the a k is a nonpositive integer.) …
14: 15.6 Integral Representations
In (15.6.2) the point 1 / z lies outside the integration contour, t b 1 and ( t 1 ) c b 1 assume their principal values where the contour cuts the interval ( 1 , ) , and ( 1 z t ) a = 1 at t = 0 . …
15: 10.25 Definitions
§10.25(ii) Standard Solutions
In particular, the principal branch of I ν ( z ) is defined in a similar way: it corresponds to the principal value of ( 1 2 z ) ν , is analytic in ( , 0 ] , and two-valued and discontinuous on the cut ph z = ± π . … It has a branch point at z = 0 for all ν . The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in ( , 0 ] , and two-valued and discontinuous on the cut ph z = ± π . … Except where indicated otherwise it is assumed throughout the DLMF that the symbols I ν ( z ) and K ν ( z ) denote the principal values of these functions. …
16: 3.11 Approximation Techniques
A sufficient condition for p n ( x ) to be the minimax polynomial is that | ϵ n ( x ) | attains its maximum at n + 2 distinct points in [ a , b ] and ϵ n ( x ) changes sign at these consecutive maxima. … There exists a unique solution of this minimax problem and there are at least k + + 2 values x j , a x 0 < x 1 < < x k + + 1 b , such that m j = m , where … is called a Padé approximant at zero of f if … A general procedure is to approximate F by a rational function R (vanishing at infinity) and then approximate f by r = 1 R . … Given n + 1 distinct points x k in the real interval [ a , b ] , with ( a = ) x 0 < x 1 < < x n 1 < x n ( = b ), on each subinterval [ x k , x k + 1 ] , k = 0 , 1 , , n 1 , a low-degree polynomial is defined with coefficients determined by, for example, values f k and f k of a function f and its derivative at the nodes x k and x k + 1 . …
17: 5.9 Integral Representations
(The fractional powers have their principal values.) … where the contour begins at , circles the origin once in the positive direction, and returns to . t z has its principal value where t crosses the positive real axis, and is continuous. … where | ph z | < π / 2 and the inverse tangent has its principal value. …
5.9.18 γ = 0 e t ln t d t = 0 ( 1 1 + t e t ) d t t = 0 1 ( 1 e t ) d t t 1 e t d t t = 0 ( e t 1 e t e t t ) d t .
18: 10.2 Definitions
This differential equation has a regular singularity at z = 0 with indices ± ν , and an irregular singularity at z = of rank 1 ; compare §§2.7(i) and 2.7(ii). … The principal branch of J ν ( z ) corresponds to the principal value of ( 1 2 z ) ν 4.2(iv)) and is analytic in the z -plane cut along the interval ( , 0 ] . … Each solution has a branch point at z = 0 for all ν . The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the z -plane along the interval ( , 0 ] . …
19: 6.16 Mathematical Applications
Hence, if x is fixed and n , then S n ( x ) 1 4 π , 0 , or 1 4 π according as 0 < x < π , x = 0 , or π < x < 0 ; compare (6.2.14). … The first maximum of 1 2 Si ( x ) for positive x occurs at x = π and equals ( 1.1789 ) × 1 4 π ; compare Figure 6.3.2. Hence if x = π / ( 2 n ) and n , then the limiting value of S n ( x ) overshoots 1 4 π by approximately 18%. Similarly if x = π / n , then the limiting value of S n ( x ) undershoots 1 4 π by approximately 10%, and so on. …
See accompanying text
Figure 6.16.2: The logarithmic integral li ( x ) , together with vertical bars indicating the value of π ( x ) for x = 10 , 20 , , 1000 . Magnify
20: 13.2 Definitions and Basic Properties
This equation has a regular singularity at the origin with indices 0 and 1 b , and an irregular singularity at infinity of rank one. …In effect, the regular singularities of the hypergeometric differential equation at b and coalesce into an irregular singularity at . … In general, U ( a , b , z ) has a branch point at z = 0 . The principal branch corresponds to the principal value of z a in (13.2.6), and has a cut in the z -plane along the interval ( , 0 ] ; compare §4.2(i). …