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This Mathematical Formaulae handbook has been prepared in response to a request from the Physics ConsultativeCommittee, with the hope that it will be useful to those studying physics. It is to some extent modelled on a similardocument issued by the Department of Engineering, but obviously reects the particular interests of physicists.There was discussion as to whether it should also include physical formulae such as Maxwells equations, etc., buta decision was taken against this, partly on the grounds that the book would become unduly bulky , but mainlybecause, in its present form, clean copies can be made available to candidates in exams.
Mathematical Formula Handbook 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 x 1 , . . . , x n ; 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 T E X 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 × 10 8 m s −1 (by definition) permeability of a vacuum µ 0 4π × 10 −7 H m −1 (by definition) permittivity of a vacuum 0 1/ µ 0 c 2 = 8·854 187 817 . . . × 10 −12 F m −1 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 A 6·022 136 7(36) × 10 23 mol −1 unified atomic mass constant m u 1·660 540 2(10) × 10 −27 kg mass of electron m e 9·109 389 7(54) × 10 −31 kg mass of proton m p 1·672 623 1(10) × 10 −27 kg Bohr magneton eh/4πm e µ B 9·274 015 4(31) × 10 −24 J T −1 molar gas constant R 8·314 510(70) J K −1 mol −1 Boltzmann constant k B 1·380 658(12) × 10 −23 J K −1 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 m 2 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 A.P. S n = a + (a + d) + (a + 2d) + ···+ [a + (n − 1)d] = n 2 [2a + (n − 1)d] G.P. S n = a + ar + ar 2 + ···+ ar n−1 = a 1 −r n 1 −r , S ∞ = a 1 − r for |r| < 1 (These results also hold for complex series.) Convergence of series: the ratio test S n = u 1 + u 2 + u 3 + ···+ u n converges as n → ∞ if lim n→∞ u n+1 u n < 1 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 (1 + x) n = 1 + nx + n(n − 1) 2! x 2 + n(n − 1)(n − 2) 3! x 3 + ··· If n is a positive integer the series terminates and is valid for all x: the term in x r is n C r x r or n r where n C r ≡ n! r!(n − r)! is the number of different ways in which an unordered sample of r objects can be selected from a set of n objects without replacement. When n is not a positive integer, the series does not terminate: the infinite series is convergent for | x | < 1. Taylor and Maclaurin Series If y(x) is well-behaved in the vicinity of x = a then it has a Taylor series, y(x) = y(a + u) = y(a) + u dy dx + u 2 2! d 2 y dx 2 + u 3 3! d 3 y dx 3 + ··· where u = x − a and the differential coefficients are evaluated at x = a. A Maclaurin series is a Taylor series with a = 0, y(x) = y(0) + x dy dx + x 2 2! d 2 y dx 2 + x 3 3! d 3 y dx 3 + ··· Power series with real variables e x = 1 + x + x 2 2! + ···+ x n n! + ··· valid for all x ln(1 + x) = x − x 2 2 + x 3 3 + ···+ (−1) n+1 x n n + ··· valid for −1 < x ≤ 1 cos x = e ix + e −ix 2 = 1 − x 2 2! + x 4 4! − x 6 6! + ··· valid for all values of x sin x = e ix − e −ix 2i = x − x 3 3! + x 5 5! + ··· valid for all values of x tan x = x + 1 3 x 3 + 2 15 x 5 + ··· valid for − π 2 < x < π 2 tan −1 x = x − x 3 3 + x 5 5 − ··· valid for −1 ≤ x ≤ 1 sin −1 x = x + 1 2 x 3 3 + 1.3 2.4 x 5 5 + ··· valid for −1 < x < 1 2 Integer series N ∑ 1 n = 1 + 2 + 3 + ···+ N = N(N + 1) 2 N ∑ 1 n 2 = 1 2 + 2 2 + 3 2 + ···+ N 2 = N(N + 1)(2N + 1) 6 N ∑ 1 n 3 = 1 3 + 2 3 + 3 3 + ···+ N 3 = [1 + 2 + 3 + ··· N] 2 = N 2 (N + 1) 2 4 ∞ ∑ 1 (−1) n+1 n = 1 − 1 2 + 1 3 − 1 4 + ··· = ln 2 [see expansion of ln(1 + x)] ∞ ∑ 1 (−1) n+1 2n − 1 = 1 − 1 3 + 1 5 − 1 7 + ··· = π 4 [see expansion of tan −1 x] ∞ ∑ 1 1 n 2 = 1 + 1 4 + 1 9 + 1 16 + ··· = π 2 6 N ∑ 1 n(n + 1)(n + 2) = 1.2.3 + 2.3.4 + ··· + N(N + 1)(N + 2) = N(N + 1)(N + 2)(N + 3) 4 This last result is a special case of the more general formula, N ∑ 1 n(n + 1)(n + 2) . . . (n + r) = N(N + 1)(N + 2) . . .(N + r)(N + r + 1) r + 2 . Plane wave expansion exp(ikz) = exp(ikr cos θ ) = ∞ ∑ l=0 (2l + 1)i l j l (kr)P l (cos θ ), where P l (cos θ ) are Legendre polynomials (see section 11) and j l (kr) are spherical Bessel functions, defined by j l ( ρ ) = π 2 ρ J l+ 1 / 2 ( ρ ), with J l (x) the Bessel function of order l (see section 11). 2. Vector Algebra If i, j, k are orthonormal vectors and A = A x i + A y j + A z k then | A | 2 = A 2 x + A 2 y + A 2 z . [Orthonormal vectors ≡ orthogonal unit vectors.] Scalar product A · B = | A || B | cos θ where θ is the angle between the vectors = A x B x + A y B y + A z B z = [ A x A y A z ] B x B y B z Scalar multiplication is commutative: A · B = B · A. Equation of a line A point r ≡ (x, y, z) lies on a line passing through a point a and parallel to vector b if r = a + λ b with λ a real number. 3 Equation of a plane A point r ≡ (x, y, z) is on a plane if either (a) r · d = | d | , where d is the normal from the origin to the plane, or (b) x X + y Y + z Z = 1 where X, Y, Z are the intercepts on the axes. Vector product A×B = n | A || B | sin θ , where θ is the angle between the vectors and n is a unit vector normal to the plane containing A and B in the direction for which A, B, n form a right-handed set of axes. A × B in determinant form i j k A x A y A z B x B y B z A × B in matrix form 0 −A z A y A z 0 −A x −A y A x 0 B x B y B z Vector multiplication is not commutative: A × B = − B × A. Scalar triple product A × B · C = A · B × C = A x A y A z B x B y B z C x C y C z = −A × C · B, etc. Vector triple product A × (B × C) = (A · C)B − (A · B)C, (A × B) × C = (A · C)B −(B · C)A Non-orthogonal basis A = A 1 e 1 + A 2 e 2 + A 3 e 3 A 1 = · A where = e 2 × e 3 e 1 · (e 2 × e 3 ) Similarly for A 2 and A 3 . Summation convention a = a i e i implies summation over i = 1 . . . 3 a · b = a i b i (a × b) i = ε i jk a j b k where ε 123 = 1; ε i jk = − ε ik j ε i jk ε klm = δ il δ jm − δ im δ jl 4 3. Matrix Algebra Unit matrices The unit matrix I of order n is a square matrix with all diagonal elements equal to one and all off-diagonal elements zero, i.e., (I) i j = δ i j . If A is a square matrix of order n, then AI = IA = A. Also I = I −1 . I is sometimes written as I n if the order needs to be stated explicitly. Products If A is a (n × l) matrix and B is a (l × m) then the product AB is defined by (AB) i j = l ∑ k=1 A ik B k j In general AB = BA. Transpose matrices If A is a matrix, then transpose matrix A T is such that (A T ) i j = (A) ji . Inverse matrices If A is a square matrix with non-zero determinant, then its inverse A −1 is such that AA −1 = A −1 A = I. (A −1 ) i j = transpose of cofactor of A i j | A | where the cofactor of A i j is (−1) i+ j times the determinant of the matrix A with the j-th row and i-th column deleted. Determinants If A is a square matrix then the determinant of A, | A | (≡ det A) is defined by | A | = ∑ i, j,k, i jk A 1i A 2 j A 3k . . . where the number of the suffixes is equal to the order of the matrix. 2×2 matrices If A = a b c d then, | A | = ad − bc A T = a c b d A −1 = 1 | A | d −b −c a Product rules (AB . . . N) T = N T . . . B T A T (AB . . . N) −1 = N −1 . . . B −1 A −1 (if individual inverses exist) | AB . . . N | = | A || B | . . . | N | (if individual matrices are square) Orthogonal matrices An orthogonal matrix Q is a square matrix whose columns q i form a set of orthonormal vectors. For any orthogonal matrix Q, Q −1 = Q T , | Q | = ±1, Q T is also orthogonal. 5 Solving sets of linear simultaneous equations If A is square then Ax = b has a unique solution x = A −1 b if A −1 exists, i.e., if | A | = 0. If A is square then Ax = 0 has a non-trivial solution if and only if | A | = 0. An over-constrained set of equations Ax = b is one in which A has m rows and n columns, where m (the number of equations) is greater than n (the number of variables). The best solution x (in the sense that it minimizes the error | Ax − b | ) is the solution of the n equations A T Ax = A T b. If the columns of A are orthonormal vectors then x = A T b. Hermitian matrices The Hermitian conjugate of A is A † = (A ∗ ) T , where A ∗ is a matrix each of whose components is the complex conjugate of the corresponding components of A. If A = A † then A is called a Hermitian matrix. Eigenvalues and eigenvectors The n eigenvalues λ i and eigenvectors u i of an n × n matrix A are the solutions of the equation Au = λ u. The eigenvalues are the zeros of the polynomial of degree n, P n ( λ ) = | A − λ I | . If A is Hermitian then the eigenvalues λ i are real and the eigenvectors u i are mutually orthogonal. | A − λ I | = 0 is called the characteristic equation of the matrix A. Tr A = ∑ i λ i , also | A | = ∏ i λ i . If S is a symmetric matrix, Λ is the diagonal matrix whose diagonal elements are the eigenvalues of S, and U is the matrix whose columns are the normalized eigenvectors of A, then U T SU = Λ and S = U Λ U T . If x is an approximation to an eigenvector of A then x T Ax/(x T x) (Rayleigh’s quotient) is an approximation to the corresponding eigenvalue. Commutators [A, B] ≡ AB − BA [A, B] = −[B, A] [A, B] † = [B † , A † ] [A + B, C] = [A, C] + [B, C] [AB, C] = A[B, C] + [A, C]B [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0 Hermitian algebra b † = (b ∗ 1 , b ∗ 2 , . . .) Matrix form Operator form Bra-ket form Hermiticity b ∗ · A · c = (A · b) ∗ · c Z ψ ∗ O φ = Z (O ψ ) ∗ φ ψ |O| φ Eigenvalues, λ real Au i = λ (i) u i O ψ i = λ (i) ψ i O | i = λ i | i Orthogonality u i · u j = 0 Z ψ ∗ i ψ j = 0 i|j = 0 (i = j) Completeness b = ∑ i u i (u i · b) φ = ∑ i ψ i Z ψ ∗ i φ φ = ∑ i | i i| φ Rayleigh–Ritz Lowest eigenvalue λ 0 ≤ b ∗ · A · b b ∗ · b λ 0 ≤ Z ψ ∗ O ψ Z ψ ∗ ψ ψ |O| ψ ψ | ψ 6 Pauli spin matrices σ x = 0 1 1 0 , σ y = 0 −i i 0 , σ z = 1 0 0 −1 σ x σ y = i σ z , σ y σ z = i σ x , σ z σ x = i σ y , σ x σ x = σ y σ y = σ z σ z = I 4. Vector Calculus Notation φ is a scalar function of a set of position coordinates. In Cartesian coordinates φ = φ (x, y, z); in cylindrical polar coordinates φ = φ ( ρ , ϕ , z); in spherical polar coordinates φ = φ (r, θ , ϕ ); in cases with radial symmetry φ = φ (r). A is a vector function whose components are scalar functions of the position coordinates: in Cartesian coordinates A = iA x + jA y + kA z , where A x , A y , A z are independent functions of x, y, z. In Cartesian coordinates ∇ (‘del’) ≡ i ∂ ∂x + j ∂ ∂y + k ∂ ∂z ≡ ∂ ∂x ∂ ∂y ∂ ∂z grad φ = ∇ φ , div A = ∇ · A, curl A = ∇ × A Identities grad( φ 1 + φ 2 ) ≡ grad φ 1 + grad φ 2 div(A 1 + A 2 ) ≡ div A 1 + div A 2 grad( φ 1 φ 2 ) ≡ φ 1 grad φ 2 + φ 2 grad φ 1 curl(A + A ) ≡ curl A 1 + curl A 2 div( φ A) ≡ φ div A + (grad φ ) · A, curl( φ A) ≡ φ curl A + (grad φ ) × A div(A 1 × A 2 ) ≡ A 2 · curl A 1 − A 1 · curl A 2 curl(A 1 × A 2 ) ≡ A 1 div A 2 − A 2 div A 1 + (A 2 · grad)A 1 − (A 1 · grad)A 2 div(curl A) ≡ 0, curl(grad φ ) ≡ 0 curl(curl A) ≡ grad(div A) − div(grad A) ≡ grad(div A) − ∇ 2 A grad(A 1 · A 2 ) ≡ A 1 × (curl A 2 ) + (A 1 · grad)A 2 + A 2 × (curl A 1 ) + (A 2 · grad)A 1 7 Grad, Div, Curl and the Laplacian Cartesian Coordinates Cylindrical Coordinates Spherical Coordinates Conversion to Cartesian Coordinates x = ρ cos ϕ y = ρ sin ϕ z = z x = r cos ϕ sin θ y = r sin ϕ sin θ z = r cos θ Vector A A x i + A y j + A z k A ρ ρ + A ϕ ϕ + A z z A r r + A θ θ + A ϕ ϕ Gradient ∇ φ ∂ φ ∂x i + ∂ φ ∂y j + ∂ φ ∂z k ∂ φ ∂ ρ ρ + 1 ρ ∂ φ ∂ ϕ ϕ + ∂ φ ∂z z ∂ φ ∂r r + 1 r ∂ φ ∂ θ θ + 1 r sin θ ∂ φ ∂ ϕ ϕ Divergence ∇ · A ∂A x ∂x + ∂A y ∂y + ∂A z ∂z 1 ρ ∂( ρ A ρ ) ∂ ρ + 1 ρ ∂A ϕ ∂ ϕ + ∂A z ∂z 1 r 2 ∂(r 2 A r ) ∂r + 1 r sin θ ∂A θ sin θ ∂ θ + 1 r sin θ ∂A ϕ ∂ ϕ Curl ∇ × A i j k ∂ ∂x ∂ ∂y ∂ ∂z A x A y A z 1 ρ ρ ϕ 1 ρ z ∂ ∂ ρ ∂ ∂ ϕ ∂ ∂z A ρ ρ A ϕ A z 1 r 2 sin θ r 1 r sin θ θ 1 r ϕ ∂ ∂r ∂ ∂ θ ∂ ∂ ϕ A r rA θ rA ϕ sin θ Laplacian ∇ 2 φ ∂ 2 φ ∂x 2 + ∂ 2 φ ∂y 2 + ∂ 2 φ ∂z 2 1 ρ ∂ ∂ ρ ρ ∂ φ ∂ ρ + 1 ρ 2 ∂ 2 φ ∂ ϕ 2 + ∂ 2 φ ∂z 2 1 r 2 ∂ ∂r r 2 ∂ φ ∂r + 1 r 2 sin θ ∂ ∂ θ sin θ ∂ φ ∂ θ + 1 r 2 sin 2 θ ∂ 2 φ ∂ ϕ 2 Transformation of integrals L = the distance along some curve ‘C’ in space and is measured from some fixed point. S = a surface area τ = a volume contained by a specified surface t = the unit tangent to C at the point P n = the unit outward pointing normal A = some vector function dL = the vector element of curve (= t dL) dS = the vector element of surface (= n dS) Then Z C A · t dL = Z C A · dL and when A = ∇ φ Z C (∇ φ ) · dL = Z C d φ Gauss’s Theorem (Divergence Theorem) When S defines a closed region having a volume τ Z τ (∇ · A) d τ = Z S (A · n) dS = Z S A · dS also Z τ (∇ φ ) d τ = Z S φ dS Z τ (∇ × A) d τ = Z S ( n × A) dS 8 [...]... ) = 2π Z ∞ −∞ exp[iω(t − τ )] dω If f (t) is an arbitrary function of t then δ (t) = 0 if t = 0, also Z ∞ −∞ Z ∞ −∞ δ (t − τ ) f (t) dt = f (τ ) δ (t) dt = 1 Reduction formulae Factorials n! = n(n − 1)(n − 2) 1, 0! = 1 Stirling’s formula for large n: For any p > −1, Z ∞ 0 For any p, q > −1, Z x p e− x dx = p 1 0 ln(n!) ≈ n ln n − n Z ∞ 0 x p (1 − x)q dx = x p−1 e− x dx = p! (− 1/2)! = √ π, ( 1/2)!... points z = ±i tan−1 z =z− n(n − 1) 2 n(n − 1)(n − 2) 3 z + z + ··· 2! 3! This last series converges both on and within the circle | z| = 1 except at the point z = −1 (1 + z)n = 1 + nz + 9 6 Trigonometric Formulae cos2 A + sin 2 A = 1 sec2 A − tan2 A = 1 cos 2A = cos 2 A − sin 2 A sin 2A = 2 sin A cos A cosec2 A − cot2 A = 1 2 tan A tan 2A = 1 − tan2 A sin ( A ± B) = sin A cos B ± cos A sin B cos A cos... m Series form of Bessel functions of the first kind (−1)k ( x/2)m+2k k!(m + k)! k=0 ∞ Jm ( x ) = ∑ (integer m) The same general form holds for non-integer m > 0 16 1 2l l! d dx l l x2 − 1 , Rodrigues’ formula so Laplace’s equation 2 u=0 If expressed in two-dimensional polar coordinates (see section 4), a solution is u(ρ, ϕ) = Aρn + Bρ−n C exp(inϕ) + D exp(−inϕ) where A, B, C, D are constants and n is... area A in the u, v plane then Z A f ( x, y) dx dy = Z A f (u, v) J du dv where The Jacobian J is also written as Z V f ( x, y, z) dx dy dz = Z V ∂x ∂u J= ∂y ∂u ∂x ∂v ∂y ∂v ∂( x, y) The corresponding formula for volume integrals is ∂(u, v) f (u, v, w) J du dv dw where now ∂x ∂u ∂y J= ∂u ∂z ∂u ∂x ∂v ∂y ∂v ∂z ∂v ∂x ∂w ∂y ∂w ∂z ∂w 14 Fourier Series and Transforms Fourier series If y( x) is a function... eimx , Cm = 1 2π Z π y( x) e−imx dx −π with m taking all integer values in the range ± M This approximation converges to y( x) as M → ∞ under the same conditions as the real form For other ranges the formulae are: Variable t, range 0 ≤ t ≤ T, frequency ω = 2π/ T, ∞ y(t) = ∑ Cm e imω t −∞ , ω Cm = 2π Variable x , range 0 ≤ x ≤ L, ∞ y( x ) = ∑ Cm e i2mπx / L , −∞ Z T 0 1 Cm = L y(t) e−imωt dt Z L 0 y(... δyn+1/2 − δyn−1/2 Approximating to derivatives dy dx d2 y dx2 n ≈ n ≈ δy 1 + δyn− 1/2 yn+1 − yn yn − yn−1 ≈ ≈ n+ /2 h h 2h where h = xn+1 − xn δ2 y n yn+1 − 2yn + yn−1 = 2 h2 h Interpolation: Everett’s formula y( x) = y( x0 + θ h) ≈ θ y0 + θ y1 + 1 1 2 θ (θ − 1)δ2 y0 + θ (θ 2 − 1)δ2 y1 + · · · 3! 3! where θ is the fraction of the interval h (= x n+1 − xn ) between the sampling points and θ = 1 − θ The... 2n) of equal sub-intervals, each of width h = (b − a)/2n; then Z 24 b a f ( x) dx ≈ h f ( a) + 4 f ( x1 ) + 2 f ( x2 ) + 4 f ( x3 ) + · · · + 2 f ( x2n−2 ) + 4 f ( x2n−1 ) + f (b) 3 Gauss’s integration formulae These have the general form For n = 2 : For n = 3 : xi = ±0·5773; Z 1 −1 n y( x) dx ≈ ∑ ci y( xi ) 1 c i = 1, 1 (exact for any cubic) xi = −0·7746, 0·0, 0·7746; c i = 0·555, 0·888, 0·555 (exact . 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,. Mathematical Formula Handbook Contents Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . 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,