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Solving sets of linear simultaneous equations If Ais square then Ax = b has a unique solution x = A“bif Av! exists, Le, if |A| #0. If Ais square then Ax = 0 has a non-trivial solution if and only if |A| = 0. An over-constrained set of equations Ax = 6 is one in which A has» rows and » 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 |x — bj) is the solution of the 1 equations A‘ Ax = Ab. If the columns of A are orthonormal vectors then x= Alb. Hermitian matrices The Hermitian conjugate of A is Al = (A°)", where A’ is a matrix each of whose components is the complex conjugate of the corresponding components of A. If A = Al then A is called a Hermitian matrix. Eigenvalues and eigenvectors The # eigenvalues A; and eigenvectors a; of an. x 1 matrix A are the solutions of the equation An = Au. The eigenvalues are the zeros of the polynomial of degree 1, P(A) = |A — Al). If A is Hermitian then the eigenvalues Ajare real and the eigenvectors 1, are mutually orthogonal. |A — Al| = 0 is called the characteristic equation of the matrix A. TrA=¥A, alsojA| =[]A). j i If S is. a symmetric matrix, A is the diagonal matrix whose diagonal elements are the eigenvalues of $, and U is the matrix whose columns are the normalized eigenvectors of A, then uUTSsU=A and S=UAUT. If x is an approximation to an eigenvector of A then x’ Ax/(x!'x) (Rayleigh’s quotient) is an approximation to the corresponding eigenvalue. Commutators [4.8] =AB-BA [A.B] =—[B,A] [4,B]t = [Bt af] [A +B,C]=[A,C] + [B,C] [ABC] = A[B,C]+[A,C]B (4, [B, C)]+ [B, [C. Al} + [€. [A,B] =0 Hermitian algebra bt = (by, b3,...) Matrix form ‘Operator form Bra-ket form Hermiticity bY Ave =(A-b) +e [woo- fore (wl0id) Eigenvalues, A real Aurj = Agar) Op, = Aj Off) =A) li) Orthogonality wy) =0 [vivi=0 tip=0 ez) Completeness b= Eailei-2) o=Eu (/v#) e=EW le Rayleigh-Ritz [wow (wloiw) Lowest eigenvalue Ao = [ew “(wI) 3. Matrix Algebra Unit matrices The unit matrix I of order 1 is a square matrix with all diagonal elements equal to one and all off-diagonal elements: zero, Le. (1)jj = 3). WA is a square matrix of order n, then Al = [A = A. Also | = r. Jis sometimes written as I, if the order needs to be stated explicitly. Products If Aisa (a x 1) matrix and Bisa (1 x mr) then the product AB is defined by L (AB) =D AaB, ket In general AB # BA. Transpose matrices If Ais a matrix, then transpose matrix AT is such that (AT)ij) = (A)ji Inverse matrices If Ais a square matrix with non-zero determinant, then its inverse A~’ is such that AA“ = A~1A = I. transpose of cofactor of Aj {Al where the cofactor of Aj is (—1)'*! times the determinant of the matrix A with the j-th row and i-th column deleted. (A i= Determinants If Ais a square matrix then the determinant of A, |Aj (= det A) is defined by [Al = 3 cia AtiArAw... ibd. where the number of the suffixes is equal to the order of the matrix. 2x2 matrices ab ta=(¢ i) then, =a raf#¢ a,tfd =b [Al =ad — be T= (h A =a’ a Product rules (AB...N)™ = NT. .BTAT (AB...) = NO, BoA (if individual inverses exist) {AB...N| =|A| |B]... [NI] (if individual matrices are square) Orthogonal matrices An orthogonal matrix Q is a square matrix whose columns q) form a set of orthonormal vectors. For any orthogonal matrix Q, Q@'=Q7, |Q|=+1, QT isalso orthogonal. Fourier transforms I y(*) is. a function defined in the range —oo < x < co then the Fourier transform (uw) is defined by the equations v(t) = sf Hwje dw, — Fw) =f ane dt. Ifwis replaced by 27f, where f is the frequency, this relationship becomes v= fainem'ap, GY) = fie er dt. If y(£) is symmetric about ¢ = 0 then v(t) = + [* G(w) cost dw, (a) =2 [ v(t) cosu ai. Tf y(t) is anti-symmetric about ! = 0 then iPlay = w(t) = = fie) sina! do, Tw) =2f y(t) sineut dt. Specific cases x a a t eo — +7 yH=a, lilst| ‘ a Sincere ' =0, |i] =a (Top Hat’), i(w) = 2a—— = 2arsine(wr) where sine(x) = y S. t “ <7: +T y(t}=a(1—|il/r), Its 3} (Sarw-tooth’), ww) = 2G = cos wr) =arsinc? (=) arr Z Co =0, lth >+. uv g JS t a y(t) =exp(-F/4) (Ganessian), W(ce) = tay Fexp (—wli/4) u(t) = (0) &" (mostutated fimetion), iw) = f(w—ay) wi) = ¥ (t= or) (sampling function) Ww) = = S(w — Jan ft) sin(x) x 21 Integer series N Xu =142434---4N—= MINED 1 $a ait testy. ant MNS RN 6 1 N ra 2 Del aE PEP pena [teasay. np MINE 1 py SM rade b adenine [see expansion of In(1+x)] 1 i a pe a eee a a Lot =1 gts 7+ ri [see expansion of tan™! x] ei 1,1,1 cal Per ttatetet == N Dole + Yu $2) 12342344 ---4 NINE T)(N-+2) = NINN BNF 8) 4 ‘This last result is a special case of the more general formula, 2. _ M(N-+1)(N-+2)...(N 4 r)(N 4 +1) SoG +101 +2)...(a +e = aCe \ T Plane wave expansion exp(ikz) = exp(ikrcos@) = Far + 1)i! ji(dr)Py(cos 8), tea where P}(cos @) are Legendre polynomials (see section 11) and j)(kr) are spherical Bessel functions, defined by Alp) = Bhewlo with J) (x) the Bessel function of order / (see section 11), 2. Vector Algebra If i, j, k are orthonormal vectors and A = A,i + Ayj + A:k then |Al? = A? + Aj + AZ [Orthonarmal vectors = orthogonal unit vectors] Scalar product A-B=|A||B\cos@ where @ is ther ac-g!e between the vector: B, = A,B, + AyBy + A.B. = [ArAyA; | Bi B. Sealar multiplication is commutative: A. B= B- A, Equation of a line A point r = (x, y,2) lies oma line passing through a point a and parallel to vector b if r=a+Ab with a real number. 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, LA., Handbook of Mathematical Functions, Dover, 1965, (Gradshteyn, LS. & Ryzhik, LM., 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 92458 x 10° ms! (by definition) permeability of a vacuum fy 4n «10-7 Hm7! (by definition) permittivity of a vacuum & — 1/poc® = 8-854 187 817... x 10-12 Fm! elementary charge e 1.602 177 33(49) x 10-1 ¢ Planck constant ht 6-626 075 5(40) = 10-4 Js h/2n th 1.054 572 66(63) x 107-4 Js Avogadro constant Nx 6-022 136 7(36) x 1073 mol! unified atomic mass constant My 1-660 540 2(10) x 10-77 kg mass of electron Me 9-109 389 7(54) x 107-9) ay mass of proton iy 1-672 623 1(10) x 10727 ke Bohr magneton efi /4rmi, Ms —-9-274.015 4(31) x 10-24 T? molar gas constant R 8-314 510(70) }K~! mol! Boltzmann constant kes 1.380 658(12) x 10-23] K-1 Stefan—Boltzmann constant o 5-670 51(19) x 10-? Wm? K-74 gravitational constant G 6-672 59(85) x 107! Nm? kg = Oller data acceleration of free fall g 9.806 65 ms~2 (standard value at sea level) 7. Hyperbolic Functions 2 4 cosh = Fle be“)ale + te valid for all x 1 ee sinhy = 5(e!— ea e+ 5 ‘ valid for all x cosh it = cos x cos ix = cosh x sinhix=isinx sinix = isinhy tanhy = sinh: sechx = — cosh x cosh.x cosh x = =. i cothr = Saha cosech.x = Sake cosh? x = sinh? x = 1 For large positive x: cosh x = sinh x — = a" _ tanh ars | = For large negative x: > ed) el cosh = —sinhx = 5 sinh x ihe 4 Relations of the functions sinhy =—sinh(—x} sechx = sech(—x) coshxy =cosh(—x) cosechx = —cosech(—x) tanhx = —tanh(—x) cothx =—coth(—x) 2 tanh (x/2) fanhx 1 + tanh? (x/2) 1 shh = —__ = cshz=-—_—__ = 1=tanh*(x/2) 4 tanh? T—tanh"(x/2) 4/1 — tanh? tanhy = 1 —sech?x sechy = f1—tanh"x cothx = Veosech?x +1 cosechx = \/coth? x—1 sinh(x/2) = Ga cosh(x/2) = yo coshx —1 sinhx tah(s/2) = sinhx coshx+1 sinh(2x) =2 sinh xcosh x tanhfeete 1+tanh’ x cosh(2r) = cosh* x + sinh? x = 2cosh? x — 1 = 1+ 2sinh?x sinh(3x) = 3sinhx +4 sinh? x cosh3x = 4cosh* x —3coshx 3 nn) = Ss 11 Grad, Div, Curl and the Laplacian ‘Cartesian Coordinates: Cylindrical Coordinates Spherical Coordinates Conversion to = = Cartesian x= poosp yxpsing saz) FTTH PSNO ye rsinesind Coordinates ~ Vector A Aud + Ayf+ Ak Aji + Ay@+ AZ A+ Aud + Agia 20, , 2, , 2 2, 18bs | %, 185, 1 eb, Gmdlentiy® || geal de” tp’ poe! oe hr Trade ranbie 1 ara.) 1 @Agsine Divergence | @Ay | @Ay 2A, | 1éleAy) | Ley | ea rand VeA ox ay oz p ép pep 1 tAy rsin@ ep 1 x be Loos te Le ci k = -= ae = @ d 2 po? 3 Pind” rane” Fi Curl Vx A a Bn j@ eB é é é es ép tp é F fo ip Ay Ay Az Ap Pay Az A, rAg rAgsing ac (r3) oe (sino5S) Laplacian Fb Fh Hp 18 (Zt) leo eb Parl é Psind 00 eo vp ot ay a | pep ap) Pog et ye te F sin? a ey" Transformation of integrals L = the distance along some curve ‘C’ in space and is measured from some fixed point. S =a surface area + =a volume contained by a specified surface # = the unit tangent to C at the point P it = the unit outward pointing normal A -=some vector function dL = the vector element of curve (=f dL) dS = the vector element of surface (= dS) Then [ A-tdt =f A-dt and when A = Vo [(ve)-at= [ao Gauss’s Theorem (Divergence Theorem) When § defines a closed region having a volume + [(v-Ajde= [ (A a) ds= f'A-as also “ & [(vear= [eas [io A)dr= [ix yas = 1 4 [aa ==rieosec ye [> costs?) ax = [" sin) ax= 55 fe exp(—x7/207) dx = ov2n J stexp(—¥?/202) dx = { fie Jains dx =—cosx+e [unex =-In(cosx) +e J cosecxdy = In(cosecx = cotx) +c [seex dx =In(secx+tanx) +c feos dx =In(sinx) +c 7 z sin(m —n)x [Snmxsinnx d= pine =a) Txdx Sx» os(n— Vat Von [sinks dx =coshx+e [cosh dy =sinhx +e [vane dx = =In{coshx)+¢ [cosechs dy = In [tanh(x/2)| +e [sechrds =2tan(e") +c [eoths dx = In(sinhx) +c sin(m + m)x 2(m—n) ~ 2(m +n) _ sin(mr—a)x | sin(m + nx fcosmscosix d= SpA ce Standard substitutions If the integrand is a function of: substitute: (@? — 7) or Vi? — x? (7 4a?) or yx? 4 a? (7 — a7) or Vx? — a? x=asinforx=acosé x= mtan@orx =asinhé x =asec@orx =acoshé forp <1 for > 2andeven forn > land odd if ne? Aa? ifer An? If the integrand is a rational function of sin.x or cos.x or both, substitute } = tan(x/2) and use the results; atten wee eons +f 1L+F 14" If the integrand is of the form: substitute: —_— x+q=ie (ax +6) /px eg F pal ax+ b= i / dx (ax b)y/ px? + yx er cE Equation of a plane A point r = (x, u,=) is ona plane if either fa} rsd =a], where d is the normal from the origin to the plane, or Le (by xtyt A where X, Y, 2 are the intercepts on the axes. ra Vector product AxB =n {Aj |B) sind, where dis the angle between the vectors and mis. a unit vector normal to the plane containing Aand Bin the direction for which A, B, n form a right-handed set of axes. A * Bin determinant form A Bin matrix form ij k 0 =A: Ay B, Ay Ay A: A; 0 Ay} | By B, By B; —Ay Ay 0 Be Vector multiplication is not commutative: A x B= —B x A. Scalar triple product Ay Ay Az AxB-.C=A-BxC=|B, B, 8. -AxC-B, etc. Cc, Gy C: Vector triple product Ax(BxC)=(A-C)B-(A-B)C, (Ax B)xC=(A-C)B-(B-C)A Non-orthogonal basis A =Aje) + Azer + Ares 2 Xe} Ay=e': A where ee = ———_ ey (ez ® 3) Similarly for Ay and Aj. Summation convention a = ae; implies summation over i =1...3 ab =ajby (@x b)) =e,a0 by where erg =15 ijn = —€inj £ijn€din = 8D jm — Bindi Fourier series for odd and even functions If y(x) is an odd (anti-symmetric) function [Le., y(—x) = —y(x)] defined in the range - < x < x, then only sines are required in the Fourier series and sj, = f° v(x)sinmx dx, If, inaddition, y(x) is symmetric about (2 x = 71/2, then the coefficients s,, are given by 3, = 0 (for m even), 3, = ah y(x) sin nex dx (for mt odd), IF u(x) is an evew (symmetric) function [ic., v(—x) = y(x)] defined in the range — < x < 7, then only constant and cosine terms are required in the Fourier series and cy = i/ v(x) dx, Cm = 2 [ vixieosms dx. If, in addition, y(x) is anti-symmetric about x = of then cy = Cand the coefficients ¢,, are given by ¢,, = 0 (for mr even), 4 ope Ca = = I v(x) cosmx dx (form odd). [These results also apply to Fourier series with more general ranges provided appropriate changes are made to the limits of integration.] Complex form of Fourier series If y(x) isa Function defined in the range —a < x < then M hex 1p oe (x)= ¥ Cael, Cu= ae [wale " dy at with m taking all integer values in the range +M. This approximation converges to y(x) as M — oo under the same conditions as the real form, For other ranges the formulae are: Variable t, range 0 < ¢ < T, frequency w = 27/T, Ea 7 = iat a treat vi) = LGve By =f yltyen™ di. Variable x’, range 0 < x' < L, He cic aas'h 1 ft th Ye) = 2 Cueto, Cu= 7h (x perm ay’, Discrete Fourier series If y(x) is a function defined in the range — < x < ~ which is sampled in the 2N equally spaced points x, = nx/N (n=—(N—1)...N], then ¥(Xn) = co +0) COSKy + €2.6082x, +--+ en. c08(N — 1) x, + cy cos Nxy +5; sln.x, + 525in 2, +--+ +sy_1sin(N — 1)x, + sy sin Nx, whtens: tive eoettictents.are: = sq Eyix) tin = LT yl) cosme, (m=1,...,.N—1) 1 N= 5G ¥yixy) cos Nxy Sm = HL vl) sins, (m=1,...,N—1) sn = x ¥Ey(x,) sin Nx, each summation being over the 2N sampling points x. sinh(x +: y) = sinh xcosh y + cosh xsink y cosh(x + y) = cosh rcosh y + sinh x sinh y tanhx + tanh y tanh(x ¥) = 7 janhxtanh y sinh +sinhy =2sinh (x + y)eosh (xy) sinhx —sinhy = Densh }(x+ y) sinh be y) 1+ tanh (x/2) = 1 = tanh(x/2) sinh(x + y) cosh xcosh y sinh(x + y) Sinhceiny, sinhx + coshx = tanh + tanhy = cothx + coth y= + Inverse functions cosh.x+cosh y = 2cosh his + y)cosh bx Ww) cashx—cosh y = 2sinh 3(x-+y) sinh 3 (x —y) afxe 2 ee for-o
a tanh ¥ = 5n(2**) for? a 2 \x-a fg worn ($s 3-1) ford a fete" poe forn #1 {wee = Fine! ta?) +0 i = dx=sin'' (=) +e [eee ten (0+ VP Ee) +e ls dy= Pte te [ VB ax =} [Vea E + sin (2)] +e 13 
