By Larry C. Andrews
This box consultant is designed to supply engineers and scientists with a brief reference for designated capabilities which are an important to resolving glossy engineering and physics difficulties. The capabilities handled during this booklet observe to many fields, together with electro-optics, electromagnetic conception, wave propagation, warmth conduction, quantum mechanics, chance concept, and electrical circuit conception, between many different components of software. a short assessment of those very important themes is integrated during this consultant, in addition to an advent to a few worthy engineering features reminiscent of the step functionality, rectangle functionality, and delta (impulse) functionality.
Table of Contents
- Engineering Functions
- countless sequence and fallacious Integrals
- Gamma Functions
- different services outlined through Integrals
- Orthogonal Polynomials
- Bessel Functions
- Orthogonal Series
- Hypergeometric-Type Functions
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Extra info for Field Guide to Special Functions for Engineers
G5) : Γ n+ (G6) : 1 = 0, Γ(− n) (G7) : Γ(x)Γ(1 − x) = (G8) : Γ (G9) : n = 0, 1, 2, . . 1 (2n)! π, = n 2 2 n! n = 0, 1, 2, . . n = 0, 1, 2, . . π sin π x 1 1 π +x Γ −x = 2 2 cos π x πΓ(2x) = 22x−1 Γ(x)Γ(x + 1/2) (Duplication formula) (G10) : Γ(a + 1) a , n = 0, 1, 2, . . Γ(a − n + 1) (Binomial coefficient) k (−1) n! Γ(k − n) (n − k)! , (k, n non-negative integers) (G11) : = 0≤k≤n Γ(− n) 0, k > n (G12) : x − x/n 1 = xeγ x Π∞ e n=1 1 + Γ(x) n (G13) : Γ(x) = x > − k, (G14) : Γ(x + k) , x(x + 1)(x + 2) .
P n (−1) = (−1)n , (P2) : P n (1) = 1, n = 0, 1, 2, . . n (P3) : P2n (0) = (−1) (2n)! )2 P2n+1 (0) = 0, n = 0, 1, 2, . . 1 dn [(x2 − 1)n ], n = 0, 1, 2, . . 2n n! dx n (P5) : (n + 1)P n+1 (x) − (2n + 1)xP n (x) + nP n−1 (x) = 0, (P4) : P n (x) = n = 1, 2, 3, . . (Recurrence formula) (P6) : P n+1 (x) − 2xP n (x) + P n−1 (x) − P n (x) = 0, n = 1, 2, 3, . . (P7) : P n+1 (x) − xP n (x) − (n + 1)P n (x), (P8) : n = 0, 1, 2, . . xP n (x) − P n−1 (x) − nP n (x) = 0, n = 1, 2, 3, . . (P9) : P n+1 (x) − P n−1 (x) = (2n + 1)P n (x), n = 1, 2, 3, .
N = 0, 1, 2, . . Gamma Functions 31 Riemann Zeta Function The Riemann zeta function, expressed as ∞ ζ(x) = 1 , x n=1 n x > 1, is closely related to the logarithm of the gamma function and to the polygamma functions. Although known to Euler, it was Riemann in 1859 who established most of its properties that are now important in the theory of numbers. An interesting series relation is given by ζ(x)(1 − 2− x ) = ∞ 1 . (2n − 1) x n=1 The zeta function also has the integral representation ζ(x) = ∞ 1 Γ(x) 0 t x−1 dt, et − 1 x > 1, and satisfies Riemann’s famous formula ζ(1 − x) = 21− x πx cos πx 2 Γ(x)ζ(x).
Field Guide to Special Functions for Engineers by Larry C. Andrews