►For infinite series involving logarithms and/or exponentials, see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §44), and Prudnikov et al. (1986a, Chapter 5).
►A generalized exponential function
satisfies the equations
…Its inverse is called a generalized logarithm.
It, too, is strictly increasing when , and
…
►For analytic generalized logarithms, see Kneser (1950).
…
►For see §6.2(i).
…
►Extensive compendia of indefinite and definite integrals of logarithms and exponentials include Apelblat (1983, pp. 16–47), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 107–116), Gröbner and Hofreiter (1950, pp. 52–90), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.3, 1.6, 2.3, 2.6).
To increase the region of validity of this equation, the logarithm of the
gamma function that appears on its left-hand side has been changed to
, where is the general logarithm.
Originally was used, where
is the principal branch of the logarithm.
►The Einstein functions and Planck’s radiation function are elementary combinations of exponentials, or exponentials and logarithms.
…
►For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv).
…
►Figure 4.3.2 illustrates the conformal mapping of the strip onto the whole -plane cut along the negative real axis, where and (principal value).
…
►►
…
►Figure 4.3.2: Conformal mapping of exponential and logarithm.
…
Magnify
…
►►