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;
          Powerseries 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
          Unitmatrices; Products; Transpose matrices; Inverse matrices; Determinants; 2×2 matrices;
          Productrules; 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. ComplexVariables.................................................................................... 9
          Complexnumbers; DeMoivre’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;
          Diracδ-‘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;
          Changingvariables 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;
          ComplexformofFourier 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
          Rangemethod; Combination of errors
      18. Statistics............................................................................................... 26
          MeanandVariance;Probability distributions; Weighted sums of random variables;
          Statistics of a data sample x1, . . ., xn; Regression (least squares fitting)
                                               Introduction
          This MathematicalFormaulaehandbook hasbeenpreparedin responseto arequestfromthePhysics Consultative
          Committee,withthehopethatitwillbeusefultothosestudyingphysics. Itistosomeextentmodelledonasimilar
          document issued by the Department of Engineering, but obviously reflects the particular interests of physicists.
          Therewasdiscussion asto whetherit 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
          Secretarywill 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
          DrDaveGreen,usingtheT Xtypesettingpackage.
                               E
          Version 1.5 December 2005.
          Bibliography
          Abramowitz,M.&Stegun,I.A.,HandbookofMathematicalFunctions,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., MathematicalHandbook ofFormulas andTables.
                (Schaum’sOutline Series, McGraw-Hill, 1968).
          Physical Constants
          Basedonthe“ReviewofParticleProperties”,Barnettetal.,1996,Physics Review D, 54,p1,and “TheFundamental
          Physical Constants”, Cohen & Taylor, 1997, Physics Today, BG7. (The figures in parentheses give the 1-standard-
          deviation uncertainties in the last digits.)
               speedoflight in a vacuum       c     2·997 924 58 × 108 m s−1  (by definition)
               permeability of a vacuum      µ0     4π×10−7 Hm−1 (bydefinition)
               permittivity of a vacuum      ǫ0     1/µ0c2 = 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
               Avogadroconstant               NA    6·022 136 7(36) × 1023 mol−1
               unifiedatomicmassconstant       mu    1·660 540 2(10) × 10−27 kg
               massofelectron                 me    9·109 389 7(54) × 10−31 kg
               massofproton                   mp    1·672 623 1(10) × 10−27 kg
               Bohrmagnetoneh/4πme           µB     9·274 015 4(31) × 10−24 J T−1
               molar gas constant             R     8·314 510(70) J K−1 mol−1
               Boltzmannconstant              kB    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 m2 kg−2
               Other data
               acceleration of free fall      g     9·806 65 m s−2  (standard value at sea level)
                                                                                                1
                                                           1. Series
       ArithmeticandGeometricprogressions
            A.P.  S =a+(a+d)+(a+2d)+···+[a+(n−1)d]= n[2a+(n−1)d]
                    n                                                    2
            G.P.  Sn = a + ar + ar2 + ··· + arn−1 = a1 − rn,                  S∞ =      a    for |r| < 1
                                                       1−r                             1−r
       (Theseresults also hold for complex series.)
       Convergenceofseries: theratio test
                                                                                  
                                                                             un+1
            Sn = u1 +u2 +u3 +···+un convergesas n → ∞ if                 lim       < 1
                                                                         n→∞ un 
       Convergenceofseries: thecomparison 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.
       Binomialexpansion
            (1+x)n = 1+nx+ n(n−1)x2+ n(n−1)(n−2)x3+···
                                     2!                3!
       If n is a positive integer the series terminates and is valid for all x: the term in xr is nC xr or n where nC ≡
                                                                                                  r       r             r
           n!     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
       convergentfor |x| < 1.
       Taylor and MaclaurinSeries
       If y(x) is well-behaved in the vicinity of x = a then it has a Taylor series,
                                        dy     u2 d2y    u3 d3y
            y(x) = y(a+u) = y(a)+udx + 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 d2y    x3 d3y
            y(x) = y(0)+ xdx + 2! dx2 + 3! dx3 +···
       Powerserieswithrealvariables
             x                   x2          xn
            e         =1+x+2! +···+ n! +···                                                                   valid for all x
                             x2    x3              n+1xn
            ln(1 + x) = x − 2 + 3 +···+(−1)            n +···                                         valid for −1 < x ≤ 1
            cosx      = eix + e−ix = 1 − x2 + x4 − x6 + ···                                         valid for all values of x
                             2             2!    4!   6!
            sinx      = eix − e−ix = x − x3 + x5 + ···                                              valid for all values of x
                             2i            3!    5!
            tanx      =x+1x3+ 2 x5+···                                                               valid for −π < x < π
                             3      15                                                                          2         2
                             x3    x5
            tan−1x    =x− 3 + 5 −···                                                                  valid for −1 ≤ x ≤ 1
                             1 x3   1.3 x5
            sin−1 x   =x+2 3 + 2.4 5 +···                                                             valid for −1 < x < 1
       2