solutions as trigonometric and hyperbolic functions
(0.008 seconds)
1—10 of 25 matching pages
1: 4.37 Inverse Hyperbolic Functions
§4.37 Inverse Hyperbolic Functions
… ► and have branch points at ; the other four functions have branch points at . … ►Each is two-valued on the corresponding cut(s), and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts. … ►For the corresponding results for , , and , use (4.37.7)–(4.37.9); compare §4.23(iv). … ►§4.37(vi) Interrelations
…2: 4.34 Derivatives and Differential Equations
3: 4.43 Cubic Equations
§4.43 Cubic Equations
…4: 14.19 Toroidal (or Ring) Functions
§14.19 Toroidal (or Ring) Functions
►§14.19(i) Introduction
►When , , , and solutions of (14.2.2) are known as toroidal or ring functions. … ►§14.19(iv) Sums
… ►§14.19(v) Whipple’s Formula for Toroidal Functions
…5: 28.32 Mathematical Applications
…
►
28.32.3
…
6: 4.23 Inverse Trigonometric Functions
§4.23 Inverse Trigonometric Functions
►§4.23(i) General Definitions
… ►§4.23(ii) Principal Values
… ►§4.23(iv) Logarithmic Forms
… ►§4.23(vii) Special Values and Interrelations
…7: 28.28 Integrals, Integral Representations, and Integral Equations
8: 28.20 Definitions and Basic Properties
9: 10.24 Functions of Imaginary Order
§10.24 Functions of Imaginary Order
… ►and , are linearly independent solutions of (10.24.1): … ►In consequence of (10.24.6), when is large and comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). Also, in consequence of (10.24.7)–(10.24.9), when is small either and or and comprise a numerically satisfactory pair depending whether or . …10: 14.15 Uniform Asymptotic Approximations
…
►See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials as with fixed.
…
►where the inverse trigonometric functions take their principal values (§4.23(ii)).
…The interval is mapped one-to-one to the interval , where is the (positive) solution of (14.15.21) when .
…
►The inverse hyperbolic and trigonometric functions take their principal values (§§4.23(ii), 4.37(ii)).
…
►(The inverse hyperbolic functions again take their principal values.)
…