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 cid 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 cid 14 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 cid 150 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 de cid 2 nite 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 cid 2 tting 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 re cid 3 ects 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 cid 147 ReviewofParticleProperties cid 148 Barnett et al 1996 Physics Review D 54 p1 and cid 147 TheFundamental Physical Constants cid 148 Cohen Taylor 1997 Physics Today BG7 The cid 2 gures in parentheses give the 1 standard deviation uncertainties in the last digits speed of light in a vacuum permeability of a vacuum permittivity of a vacuum elementary charge Planck constant h 2 cid 25 Avogadro constant uni cid 2 ed atomic mass constant mass of electron mass of proton Bohr magneton eh 4 cid 25 me molar gas constant Boltzmann constant Stefan cid 150 Boltzmann constant gravitational constant Other data c cid 22 0 cid 15 0 e h h NA mu me mp cid 22 B R kB cid 27 G g 108 m s cid 0 1 by de cid 2 nition by de cid 2 nition 854 187 817 10 cid 0 12 F m cid 0 1 2 997 924 58 cid 1 4 cid 25 cid 2 1 cid 22 0c2 8 10 cid 0 cid 2 7 H m cid 0 1 cid 1 602 177 33 49 626 075 5 40 054 572 66 63 022 136 7 36 660 540 2 10 109 389 7 54 672 623 1 10 1 cid 2 19 C 10 cid 0 cid 2 34 J s 10 cid 0 34 J s 10 cid 0 cid 2 1023 mol cid 0 27 kg 10 cid 0 31 kg 10 cid 0 27 kg 10 cid 0 24 J T cid 0 1 cid 2 cid 2 cid 2 cid 2 cid 2 274 015 4 31 cid 2 314 510 70 J K cid 0 380 658 12 1 10 cid 0 1 mol cid 0 23 J K cid 0 2 K cid 0 8 W m cid 0 2 11 N m2 kg cid 0 1 4 10 cid 0 cid 2 10 cid 0 10 cid 0 670 51 19 672 59 85 cid 2 cid 2 2 1 6 1 6 1 9 1 9 8 1 5 6 cid 1 cid 1 cid 1 cid 1 cid 1 cid 1 cid 1 cid 1 cid 1 cid 1 cid 1 cid 1 cid 1 acceleration of free fall 9 806 65 m s cid 0 standard value at sea level 1 1 Series Arithmetic and Geometric progressions A P Sn a a d a 2d a n 1 d 2a n 1 d n 2 cid 1 cid 1 cid 1 cid 0 rn r 1 1 cid 0 cid 0 cid 0 S1 cid 18 a cid 0 1 r for r j j 1 cid 19 G P Sn a ar ar2 arn cid 0 1 a cid 1 cid 1 cid 1 These results also hold for complex series Convergence of series the ratio test Sn u1 u2 u3 un converges as n cid 1 cid 1 cid 1 1 if lim 1 n Convergence of series the comparison test 1 un 1 un cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 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 …