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Additional resources for Answers to a Selection Problems from Classical Electrodynamics John David Jackson
T(c) T(a m) m-l (_z)-a "" (a)"T(m-n) z-n + Jarg (-z) J< 1&, JzJ > t20 n! T(c - a - n) , 1, c - a =f: 0, ±1, ±2, ... In the last two results the case e - a = 0, -1, -2, ... , becomes elementary and may be obtained directly from the differentiation formulas. The case c - a = 1,2,3, '" is obtained from the formula tz: [(1- z)a+b- c 2Fl(a, b; e; z)] = (c - a)" (c - b) .. (1 _ z)a+b-c-n F (a b' c (c)" . 4 Linear and higher order transformations by taking the limit as c ~ a. . _ r(a + b) ~ (a)" (b)" 2F 1(a, b, a + b, z) - r(a) r(b) 6& (nT)"2 + 1) - Y'(a + n) - Y'(b + n) -log (1 Iarg (1 - z) I < n, 11 - z I < 1, X [2Y'(n z)] (1 - z)", ~1(a,b;a+b+m;z) "'-1 + b + m) ~ (a)" (b)" (1 _ r(a + m) r(b + m) ,,=0 n!
Berlin: Springer 1924. TITCHMARSH, E. : The Zeta function of Riemann. Cambridge (England) 1930. WHITTAKER, E. , and G. N. WATSON: A course of modern analysis. Cambridge (England) 1927. 1 Definitions and elementary relations The function represented by the infinite series i: ,,=0 (a)(")(b),, z"l c.. * The symbol (a),. is defined as (a),. = a (a + 1) (a + 2) .. , (a + n - 1) _ r(a +n) r(a) Thus 2F .. ' The series converges in the unit circle Iz I < 1. The behavior of the series on its circle of convergence Iz I = 1 is given by a) divergent for Re (a + b - c) > 1, b) absolutely convergent for Re (a b - c) < 0, c) conditionally convergent for 0 < Re (a b - c) < 1, the point z = 1 being excluded.
Recurrence formula n-1 L (:) Bm =0, n = 2, 3, 4, ... m~O Generalized Bernoulli numbers and polynomials The generalized Bernoulli numbers B~m) and the generalized Bernoulli polynomials B~m} (x) are defined by: (m = 0, 1, 2, ... :::.. ~ -Tzn, B(rn) -Z-- n~O n. eXZ(_Z_Z_)m = i:B~m)ix) e - 1, Recurrence formulas (n - m) B~m} (x) n~O n. = nB~":21 (x) = < zn, JzJ 2n, < 2n. 1) B~":21 (x), (1 - ~) B~m} (x) + n (~ - 1) B~":21 (x), L (7) (_1)1-1 (1 - 1)! B~~ll} (x). 5 Bernoulli and Euler polynomials Some inequalities involving Bernoulli numbers and' Bernoulli polynomials IB 2n (x) 1< IB2nl, n = 1,2,3, ...