real case

SearchAdvancedHelp

(0.004 seconds)

11—20 of 224 matching pages

11: 32.11 Asymptotic Approximations for Real Variables

… ►Consider the special case of ${\text{P}}_{\text{II}}$ with $\alpha =0$: … ►Any nontrivial real solution of (32.11.12) satisfies … ►In the case when …where $k$ is a nonzero real constant. … ►In the generic case …

12: 8.2 Definitions and Basic Properties

… ►However, when the integration paths do not cross the negative real axis, and in the case of (8.2.2) exclude the origin, $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ take their principal values; compare §4.2(i). …

13: 23.4 Graphics

… ►

§23.4(i) Real Variables

… ►

►►

Figure 23.4.7: $\mathrm{\wp }\left(x\right)$ with ${\omega }_1=K\left(k\right)$, ${\omega }_3=\mathrm{i}{K}^{\prime }\left(k\right)$ for $0\le x\le 9$, $k^2$ = 0. … Magnify

…

14: 14.13 Trigonometric Expansions

… ►These Fourier series converge absolutely when . If then they converge, but, if $\theta \ne \frac{1}{2}\pi$, they do not converge absolutely. …

15: 19.20 Special Cases

… ►When the variables are real and distinct, the various cases of $R_J(x,y,z,p)$ are called circular (hyperbolic) cases if $(p-x)(p-y)(p-z)$ is positive (negative), because they typically occur in conjunction with inverse circular (hyperbolic) functions. …

16: 12.14 The Function $W(a,x)$

… ►This equation is important when $a$ and $z$ $(=x)$ are real, and we shall assume this to be the case. … ►In this case there are no real turning points, and the solutions of (12.2.3), with $z$ replaced by $x$, oscillate on the entire real $x$-axis. …

17: 10.41 Asymptotic Expansions for Large Order

… ►In the case of (10.41.13) with positive real values of $z$ the result is a consequence of the error bounds given in Olver (1997b, pp. 377–378). …

18: 19.29 Reduction of General Elliptic Integrals

… ►Cubic cases of these formulas are obtained by setting one of the factors in (19.29.3) equal to 1. … ►All other cases are integrals of the second kind. … ►In the cubic case ($h=3$) the basic integrals are …In the quartic case ($h=4$) the basic integrals are … ►A special case of Carlson (1999, (2.19)) is given by …

19: 2.8 Differential Equations with a Parameter

… ►There are three main cases. … ►for Case I, …for Case II, …for Case III. … ►In Case III the approximating equation is …

20: 18.40 Methods of Computation

… ►Given the power moments, ${\mu }_n={\int }_a^b{x}^{n}d\mu (x)$, $n=0,1,2,\mathrm{\dots }$, can these be used to find a unique $\mu (x)$, a non-decreasing, real, function of $x$, in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant. … ►In what follows we consider only the simple, illustrative, case that $\mu (x)$ is continuously differentiable so that $d\mu (x)=w(x)dx$, with $w(x)$ real, positive, and continuous on a real interval $[a,b].$ The strategy will be to: 1) use the moments to determine the recursion coefficients ${\alpha }_n,{\beta }_n$ of equations (18.2.11_5) and (18.2.11_8); then, 2) to construct the quadrature abscissas $x_i$ and weights (or Christoffel numbers) $w_i$ from the J-matrix of §3.5(vi), equations (3.5.31) and(3.5.32). …