Mathematical Formaulae handbook

Mathematical Formaulae handbook

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Introduction............................................................................................ 1 Bibliography; PhysicalConstants 1. Series.................................................................................................... 2 Arithmeticand Geometricprogressions; Convergenceof series: theratio test; Convergence ofseries: the comparisontest;Binomialexpansion; Taylorand Maclaurin Series; Power series withreal variables; Integer series; Plane waveexpansion 2. Vector Algebra......................................................................................... 3 Scalar product;Equation of aline; Equationof a plane; Vector product; Scalar tripleproduct; Vector tripleproduct;Non-orthogonalbasis;Summationconvention 3. Matrix Algebra ........................................................................................ 5 Unitmatrices;Products; Transpose matrices; Inverse matrices; Determinants; 2
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2matrices; Productrules; Orthogonalmatrices;Solvingsets oflinear simultaneous equations;Hermitian matrices; Eigenvalues andeigenvectors;Commutators;Hermitianalgebra;Pauli spin matrices 4. Vector Calculus........................................................................................ 7 Notation;Identities;Grad,Div,Curl andthe Laplacian; Transformationof integrals 5. Complex Variables.................................................................................... 9 Complexnumbers; DeMoivre’s theorem;Power series for complexvariables. 6. Trigonometric Formulae............................................................................ 10 Relations between sides and angles ofany planetriangle; Relations between sides and angles ofany sphericaltriangle 7. Hyperbolic Functions ............................................................................... 11 Relations of thefunctions; Inverse functions 8. Limits.................................................................................................. 12 9. Differentiation........................................................................................ 13 10. Integration............................................................................................ 13 Standardforms;Standardsubstitutions;Integrationby parts;Differentiationof an integral; Dirac d -‘function’;Reduction formulae 11. Differential Equations............................................................................... 16 Diffusion (conduction)equation;Wave equation;Legendre’s equation; Bessel’s equation; Laplace’s equation; Sphericalharmonics 12. Calculus of Variations............................................................................... 17 13. Functions of Several Variables..................................................................... 18 Taylorseries for twovariables;Stationarypoints;Changingvariables: the chainrule; Changingvariables in surface andvolumeintegrals – Jacobians 14. Fourier Series and Transforms..................................................................... 19 Fourier series; Fourier series for otherranges;Fourier series for oddandeven functions; Complexformof Fourier series;Discrete Fourier series; Fourier transforms;Convolution theorem; Parseval’s theorem; Fourier transforms in twodimensions; Fourier transforms in threedimensions 15. LaplaceTransforms.................................................................................. 23 16. Numerical Analysis ................................................................................. 24 Findingthezeros of equations;Numerical integrationof differential equations; Central difference notation;Approximatingtoderivatives;Interpolation: Everett’sformula; Numericalevaluation ofde?nite integrals 17. Treatment of Random Errors....................................................................... 25 Range method; Combination oferrors 18. Statistics............................................................................................... 26 Mean and Variance;Probability distributions; Weightedsums of randomvariables; Statisticsof adatasample x1,...,xn;Regression (leastsquares ?tting)
Introduction
This MathematicalFormaulae handbook has been preparedin response to a request from the Physics Consultative Committee,with thehopethatitwillbeusefultothosestudyingphysics. Itistosomeextentmodelledonasimilar document issued by the Department of Engineering, but obviously re?ects the particular interests of physicists. Therewas 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 presentform, cleancopies canbe madeavailableto candidatesin 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 Secretarywill also begratefulto be informed ofany (equally inevitable)errorswhich arefound. This book was compiled by Dr John Shakeshaft and typeset originally by Fergus Gallagher, and currently by Dr DaveGreen,using theTEX typesetting package. Version 1.5December 2005.
Bibliography Abramowitz,M.&Stegun, I.A.,Handbook of MathematicalFunctions, Dover,1965. Gradshteyn, I.S.& Ryzhik, I.M.,Tableof Integrals,SeriesandProducts, AcademicPress,1980. Jahnke,E.& Emde,F., Tablesof Functions, Dover,1986. Nordling, C.& ¨Osterman, J.,Physics Handbook, Chartwell-Bratt,Bromley,1980. Speigel,M.R.,MathematicalHandbook ofFormulas andTables. (Schaum’sOutline Series,McGraw-Hill,1968).
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