| 释义 |
Hurwitz Zeta FunctionA generalization of the Riemann Zeta Function with a Formula
 | (1) |
 is excluded. The Hurwitz zeta function can also be given by the functional equation
 | (2) |
 | (3) |
 , then
 | (4) |
where  is the Gamma Function.The Polygamma Function  can be expressed in terms of the Hurwitz zeta function by
 | (8) |
For Positive integers  ,  , and  , where  is a Bernoulli Number,  a Bernoulli Polynomial,  is a Polygamma Function, and is a Riemann Zeta Function (Miller and Adamchik). Miller and Adamchik also give the closed-form expressionsSee also Khintchine's Constant, Polygamma Function, PsiFunction, Riemann Zeta Function, Zeta Function
References
Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1995.Elizalde, E.; Odintsov, A. D.; and Romeo, A. Zeta Regularization Techniques with Applications. River Edge, NJ: World Scientific, 1994. Knopfmacher, J. ``Generalised Euler Constants.'' Proc. Edinburgh Math. Soc. 21, 25-32, 1978. Magnus, W. and Oberhettinger, F. Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. New York: Springer-Verlag, 1966. Miller, J. and Adamchik, V. ``Derivatives of the Hurwitz Zeta Function for Rational Arguments.'' Submitted to J. Symb. Comput. Spanier, J. and Oldham, K. B. ``The Hurwitz Function  .'' Ch. 62 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 653-664, 1987. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 268-269, 1950. |
