solutions as trigonometric and hyperbolic functions
(0.007 seconds)
11—20 of 25 matching pages
11: 14.20 Conical (or Mehler) Functions
§14.20(v) Trigonometric Expansion
… ►where the inverse trigonometric functions take their principal values. …12: 10.45 Functions of Imaginary Order
§10.45 Functions of Imaginary Order
… ►and , are real and linearly independent solutions of (10.45.1): … ►In consequence of (10.45.5)–(10.45.7), and comprise a numerically satisfactory pair of solutions of (10.45.1) when is large, and either and , or and , comprise a numerically satisfactory pair when is small, depending whether or . … ►In this reference is denoted by . …13: 36.11 Leading-Order Asymptotics
14: 28.10 Integral Equations
§28.10(i) Equations with Elementary Kernels
… ►§28.10(ii) Equations with Bessel-Function Kernels
… ►§28.10(iii) Further Equations
…15: 10.20 Uniform Asymptotic Expansions for Large Order
§10.20 Uniform Asymptotic Expansions for Large Order
… ►Define to be the solution of the differential equation … ►In this way there is less usage of many-valued functions. … ►where is the positive root of the equation . … ►§10.20(iii) Double Asymptotic Properties
…16: 32.2 Differential Equations
17: Bibliography L
18: 22.19 Physical Applications
19: 14.3 Definitions and Hypergeometric Representations
§14.3(iii) Alternative Hypergeometric Representations
… ►§14.3(iv) Relations to Other Functions
…20: Errata
In the first sentence of this subsection, the constraint has been replaced with .
Originally, the factor on the right-hand side was written as , which was taken directly from Watson (1944, p. 412, (13.46.5)), who uses a different normalization for the associated Legendre function of the second kind . Watson’s equals in the DLMF.
Reported by Arun Ravishankar on 2018-10-22
Scales were corrected in all figures. The interval was replaced by and replaced by . All plots and interactive visualizations were regenerated to improve image quality.
(a) Density plot. | (b) 3D plot. |
Figure 36.3.9: Modulus of hyperbolic umbilic canonical integral function .
(a) Density plot. | (b) 3D plot. |
Figure 36.3.10: Modulus of hyperbolic umbilic canonical integral function .
(a) Density plot. | (b) 3D plot. |
Figure 36.3.11: Modulus of hyperbolic umbilic canonical integral function .
(a) Density plot. | (b) 3D plot. |
Figure 36.3.12: Modulus of hyperbolic umbilic canonical integral function .
Reported 2016-09-12 by Dan Piponi.
The first paragraph has been rewritten to correct reported errors. The new version is reproduced here.
Let and be real constants and
The roots of
are:
-
(a)
, , and , with , when .
-
(b)
, , and , with , when , , and .
-
(c)
, , and , with , when .
Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. See also §1.11(iii).
Reported 2014-10-31 by Masataka Urago.
Originally the limiting form for in the last line of this table was incorrect (, instead of ).
Reported 2010-11-23.