value at T=0

(0.004 seconds)

11—14 of 14 matching pages

11: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►where

g_{k}

k=0,1,2,\dots

, are the coefficients that appear in the asymptotic expansion (5.11.3) of

\Gamma\left(z\right)

. The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at

\eta=0

, and the Maclaurin series expansion of

c_{k}(\eta)

is given by …where

d_{0,0}=-\tfrac{1}{3}

, …For numerical values of

d_{k,n}

to 30D for

k=0(1)9

and

n=0(1)N_{k}

, where

N_{k}=28-4\left\lfloor k/2\right\rfloor

, see DiDonato and Morris (1986). … ►A different type of uniform expansion with coefficients that do not possess a removable singularity at

z=a

is given by …

12: 19.36 Methods of Computation

… ►Accurate values of

F\left(\phi,k\right)-E\left(\phi,k\right)

for

k^{2}

near 0 can be obtained from

R_{D}

by (19.2.6) and (19.25.13). … ►where

n=0,1,2,\dots

, and …As

n\to\infty

c_{n}

a_{n}

, and

t_{n}

converge quadratically to limits

0

M

, and

T

, respectively; hence … ►(19.22.20) reduces to

0=0

p=x

p=y

, and (19.22.19) reduces to

0=0

z=x

z=y

. … ►This method loses significant figures in

\rho

\alpha^{2}

and

k^{2}

are nearly equal unless they are given exact values—as they can be for tables. …

13: 2.4 Contour Integrals

… ►Assume that

p(t)

and

q(t)

are analytic on an open domain

\mathbf{T}

that contains

\mathscr{P}

, with the possible exceptions of

t=a

and

t=b

. … ►Now suppose that in (2.4.10) the minimum of

\Re\left(zp(t)\right)

\mathscr{P}

occurs at an interior point

t_{0}

. …and apply the result of §2.4(iii) to each integral on the right-hand side, the role of the series (2.4.11) being played by the Taylor series of

p(t)

and

q(t)

t=t_{0}

. … ►Cases in which

p^{\prime}(t_{0})\neq 0

are usually handled by deforming the integration path in such a way that the minimum of

\Re\left(zp(t)\right)

is attained at a saddle point or at an endpoint. … ►with

p, q

and their derivatives evaluated at

t_{0}

. …

14: 26.10 Integer Partitions: Other Restrictions

… ►The set

\{2,3,4,\ldots\}

is denoted by

T

. If more than one restriction applies, then the restrictions are separated by commas, for example,

p\left(\mathcal{D}2,\hbox{}\!\!\in\!T,n\right)

. … ►where the sum is over nonnegative integer values of

k

for which

n-\frac{1}{2}(3k^{2}\pm k)\geq 0

. … ►where the sum is over nonnegative integer values of

k

for which

n-(3k^{2}\pm k)\geq 0

. … ►where the sum is over nonnegative integer values of

m

for which

n-\tfrac{1}{2}km^{2}-m+\tfrac{1}{2}km\geq 0

. …