Contents Introduction . 1 Bibliography; Physical Constants 1. Series . 2 Arithmetic and Geometric progressions; Convergence of series: the ratio test; Convergence of series: the comparison test; Binomial expansion; Taylor and Maclaurin Series; Power series with real variables; Integer series; Plane wave expansion 2. Vector Algebra . 3 Scalar product; Equation of a line; Equation of a plane; Vector product; Scalar triple product; Vector triple product; Non-orthogonal basis; Summation convention 3. Matrix Algebra . 5 Unit matrices; Products; Transpose matrices; Inverse matrices; Determinants; 2×2 matrices; Product rules; Orthogonal matrices; Solving sets of linear simultaneous equations; Hermitian matrices; Eigenvalues and eigenvectors; Commutators; Hermitian algebra; Pauli spin matrices 4. Vector Calculus . 7 Notation; Identities; Grad, Div, Curl and the Laplacian; Transformation of integrals 5. Complex Variables . 9 Complex numbers; De Moivre's theorem; Power series for complex variables. 6. Trigonometric Formulae . 10 Relations between sides and angles of any plane triangle; Relations between sides and angles of any spherical triangle 7. Hyperbolic Functions . 11 Relations of the functions; Inverse functions 8. Limits . 12 9. Differentiation . 13 10. Integration . 13 Standard forms; Standard substitutions; Integration by parts; Differentiation of an integral; Dirac δ-`function'; Reduction formulae 11. Differential Equations . 16 Diffusion (conduction) equation; Wave equation; Legendre's equation; Bessel's equation; Laplace's equation; Spherical harmonics 12. Calculus of Variations . 17 13. Functions of Several Variables . 18 Taylor series for two variables; Stationary points; Changing variables: the chain rule; Changing variables in surface and volume integrals – Jacobians 14. Fourier Series and Transforms . 19 Fourier series; Fourier series for other ranges; Fourier series for odd and even functions; Complex form of Fourier series; Discrete Fourier series; Fourier transforms; Convolution theorem; Parseval's theorem; Fourier transforms in two dimensions; Fourier transforms in three dimensions 15. Laplace Transforms . 23 16. Numerical Analysis . 24 Finding the zeros of equations; Numerical integration of differential equations; Central difference notation; Approximating to derivatives; Interpolation: Everett's formula; Numerical evaluation of definite integrals 17. Treatment of Random Errors . 25 Range method; Combination of errors 18. Statistics . 26 Mean and Variance; Probability distributions; Weighted sums of random variables; Statistics of a data sample x1, . , xn; Regression (least squares fitting) Introduction This Mathematical Formaulae handbook has been prepared in response to a request from the Physics Consultative Committee, with the hope that it will be useful to those studying physics. It is to some extent modelled on a similar document issued by the Department of Engineering, but obviously reflects the particular interests of physicists. There was discussion as to whether it should also include physical formulae such as Maxwell's equations, etc., but a decision was taken against this, partly on the grounds that the book would become unduly bulky, but mainly because, in its present form, clean copies can be made available to candidates in exams. There has been wide consultation among the staff about the contents of this document, but inevitably some users will seek in vain for a formula they feel strongly should be included. Please send suggestions for amendments to the Secretary of the Teaching Committee, and they will be considered for incorporation in the next edition. The Secretary will also be grateful to be informed of any (equally inevitable) errors which are found. This book was compiled by Dr John Shakeshaft and typeset originally by Fergus Gallagher, and currently by Dr Dave Green, using the TEX typesetting package. Version 1.5 December 2005. Bibliography Abramowitz, M. & Stegun, I.A., Handbook of Mathematical Functions, Dover, 1965. Gradshteyn, I.S. & Ryzhik, I.M., Table of Integrals, Series and Products, Academic Press, 1980. Jahnke, E. & Emde, F., Tables of Functions, Dover, 1986. Nordling, C. & Osterman,¨ J., Physics Handbook, Chartwell-Bratt, Bromley, 1980. Speigel, M.R., Mathematical Handbook of Formulas and Tables. (Schaum's Outline Series, McGraw-Hill, 1968). Physical Constants Based on the “Review of Particle Properties”, Barnett et al., 1996, Physics Review D, 54, p1, and “The Fundamental Physical Constants”, Cohen & Taylor, 1997, Physics Today, BG7. (The figures in parentheses give the 1-standard- deviation uncertainties in the last digits.) speed of light in a vacuum c 2 997 924 58 108 m s 1 (by definition) · × − permeability of a vacuum µ 4π 10 7 H m 1 (by definition) 0 × − − permittivity of a vacuum 1=µ c2 = 8 854 187 817 . 10 12 F m 1 0 0 · × − − elementary charge e 1 602 177 33(49) 10 19 C · × − Planck constant h 6 626 075 5(40) 10 34 J s · × − h=2π ¯h¯ 1 054 572 66(63) 10 34 J s · × − Avogadro constant N 6 022 136 7(36) 1023 mol 1 A · × − unified atomic mass constant m 1 660 540 2(10) 10 27 kg u · × − mass of electron m 9 109 389 7(54) 10 31 kg e · × − mass of proton m 1 672 623 1(10) 10 27 kg p · × − Bohr magneton eh=4πm µ 9 274 015 4(31) 10 24 J T 1 e B · × − − molar gas constant R 8 314 510(70) J K 1 mol 1 · − − Boltzmann constant k 1 380 658(12) 10 23 J K 1 B · × − − Stefan–Boltzmann constant σ 5 670 51(19) 10 8 W m 2 K 4 · × − − − gravitational constant G 6 672 59(85) 10 11 N m2 kg 2 · × − − Other data acceleration of free fall g 9 806 65 m s 2 (standard value at sea level) · − 1 1. Series Arithmetic and Geometric progressions n A.P. S = a + (a + d) + (a + 2d) + + [a + (n 1)d] = [2a + (n 1)d] n · · · − 2 − n 2 n 1 1 r a G.P. S = a + ar + ar + + ar − = a − , S1 = for r < 1 n · · · 1 r 1 r j j − − (These results also hold for complex series.) Convergence of series: the ratio test un+1 Sn = u1 + u2 + u3 + + un converges as n 1 if lim < 1 · · · ! n 1 u ! n Convergence of series: the comparison test If each term in a series of positive terms is less than the corresponding term in a series known to be convergent, then the given series is also convergent. Binomial expansion n(n 1) n(n 1)(n 2) (1 + x)n = 1 + nx + − x2 + − − x3 + 2! 3! · · · r n r n n If n is a positive integer the series terminates and is valid for all x: the term in x is Crx or where Cr n! r ≡ is the number of different ways in which an unordered sample of r objects can be selected from a set of r!(n r)! n objects− without replacement. When n is not a positive integer, the series does not terminate: the infinite series is convergent for x < 1. j j Taylor and Maclaurin Series If y(x) is well-behaved in the vicinity of x = a then it has a Taylor series, dy u2 d2 y u3 d3 y y(x) = y(a + u) = y(a) + u + + + dx 2! dx2 3! dx3 · · · where u = x a and the differential coefficients are evaluated at x = a. A Maclaurin series is a Taylor series with − a = 0, dy x2 d2 y x3 d3 y y(x) = y(0) + x + + + dx 2! dx2 3! dx3 · · · Power series with real variables x2 xn ex = 1 + x + + + + valid for all x 2! · · · n! · · · x2 x3 xn ln(1 + x) = x + + + ( 1)n+1 + valid for 1 < x 1 − 2 3 · · · − n · · · − ≤ ix ix 2 4 6 e + e− x x x cos x = = 1 + + valid for all values of x 2 − 2! 4! − 6! · · · ix ix 3 5 e e− x x sin x = − = x + + valid for all values of x 2i − 3! 5! · · · 1 2 π π tan x = x + x3 + x5 + valid for < x < 3 15 · · · − 2 2 3 5 1 x x tan− x = x + valid for 1 x 1 − 3 5 − · · · − ≤ ≤ 3 5 1 1 x 1.3 x sin− x = x + + + valid for 1 < x < 1 2 3 2.4 5 · · · − 2 Integer series N N(N + 1) ∑ n = 1 + 2 + 3 + + N = 1 · · · 2 N N(N + 1)(2N + 1) ∑ n2 = 12 + 22 + 32 + + N2 = 1 · · · 6 N N2(N + 1)2 ∑ n3 = 13 + 23 + 33 + + N3 = [1 + 2 + 3 + N]2 = 1 · · · · · · 4 1 ( 1)n+1 1 1 1 ∑ − = 1 + + = ln 2 [see expansion of ln(1 + x)] 1 n − 2 3 − 4 · · · 1 n+1 ( 1) 1 1 1 π 1 ∑ − = 1 + + = [see expansion of tan− x] 2n 1 − 3 5 − 7 · · · 4 1 − 1 2 ∑ 1 1 1 1 π 2 = 1 + + + + = 1 n 4 9 16 · · · 6 N N(N + 1)(N + 2)(N + 3) ∑ n(n + 1)(n + 2) = 1.2.3 + 2.3.4 + + N(N + 1)(N + 2) = 1 · · · 4 This last result is a special case of the more general formula, N N(N + 1)(N + 2) . (N + r)(N + r + 1) ∑ n(n + 1)(n + 2) . (n + r) = . 1 r + 2 Plane wave expansion 1 l exp(ikz) = exp(ikr cosθ) = ∑(2l + 1)i jl(kr)Pl(cosθ), l=0 where Pl(cosθ) are Legendre polynomials (see section 11) and jl(kr) are spherical Bessel functions, defined by π j (ρ) = J 1 (ρ), with J (x) the Bessel function of order l (see section 11). l 2ρ l+ =2 l r 2. Vector Algebra If i, j, k are orthonormal vectors and A = A i + A j + A k then A 2 = A2 + A2 + A2. [Orthonormal vectors x y z j j x y z ≡ orthogonal unit vectors.] Scalar product A B = A B cosθ where θ is the angle between the vectors · j j j j Bx = A B + A B + A B = [ Ax Ay Az ] B x x y y z z 2 y 3 Bz 4 5 Scalar multiplication is commutative: A B = B A.