as%20x%E2%86%92%C2%B11
(0.003 seconds)
21—30 of 687 matching pages
21: 23 Weierstrass Elliptic and Modular
Functions
…
22: 7.8 Inequalities
…
►Let denote Mills’ ratio:
…
►
7.8.4
,
►
7.8.5
.
…
►The function is strictly decreasing for .
For these and similar results for Dawson’s integral see Janssen (2021).
…
23: 10.3 Graphics
24: 22.3 Graphics
…
►Line graphs of the functions , , , , , , , , , , , and for representative values of real and real illustrating the near trigonometric (), and near hyperbolic () limits.
…
►
, , and as functions of real arguments and .
…
►
►
►
…
25: 5.22 Tables
…
►Abramowitz and Stegun (1964, Chapter 6) tabulates , , , and for to 10D; and for to 10D; , , , , , , , and for to 8–11S; for to 20S.
Zhang and Jin (1996, pp. 67–69 and 72) tabulates , , , , , , , and for to 8D or 8S; for to 51S.
…
►Abramov (1960) tabulates for () , () to 6D.
…This reference also includes for the same arguments to 5D.
Zhang and Jin (1996, pp. 70, 71, and 73) tabulates the real and imaginary parts of , , and for , to 8S.
26: 36.5 Stokes Sets
…
►The Stokes set consists of the rays in the complex -plane.
…
►where are the two smallest positive roots of the equation
…
►The first sheet corresponds to and is generated as a solution of Equations (36.5.6)–(36.5.9).
…
►When the Stokes set is given by
…
►Alternatively, when
…
27: Bibliography N
…
►
On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
…
►
Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
…
►
The resurgence properties of the incomplete gamma function II.
Stud. Appl. Math. 135 (1), pp. 86–116.
…
►
A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
…
►
Symmetries in the fourth Painlevé equation and Okamoto polynomials.
Nagoya Math. J. 153, pp. 53–86.
…
28: 3.4 Differentiation
…
►If is continuous on the interval defined in §3.3(i), then the remainder in (3.4.1) is given by
…
►where is a simple closed contour described in the positive rotational sense such that and its interior lie in the domain of analyticity of , and is interior to .
Taking to be a circle of radius centered at , we obtain
…
►
, .
…
►For partial derivatives we use the notation .
…
29: 7.23 Tables
…
►
•
►
•
…
►
•
►
•
…
Abramowitz and Stegun (1964, Chapter 7) includes , , , 10D; , , 8S; , , 7D; , , , 6S; , , 10D; , , 9D; , , , 7D; , , , , 15D.
Abramowitz and Stegun (1964, Table 27.6) includes the Goodwin–Staton integral , , 4D; also , , 4D.
Zhang and Jin (1996, pp. 637, 639) includes , , , 8D; , , , 8D.
§7.23(iii) Complex Variables,
… ►Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of , , , 7D and 8D, respectively; the real and imaginary parts of , , , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.
30: 36 Integrals with Coalescing Saddles
…