BasicsGoniometric functionsHyperbolic functionsCalculusLimitsComplex numbers and quaternionsComplex numbersQuaternionsGeometryTrianglesCurvesVectorsSeriesExpansionConvergence and divergence of seriesConvergence and divergence of functionsProducts and quotientsLogarithmsPolynomialsPrimesCalculusIntegralsArithmetic rulesArc lengts, surfaces and volumesSeparation of quotientsSpecial functionsGoniometric integralsFunctions with more variablesDerivativesTaylor seriesExtremaThe -operatorIntegral theoremsMultiple integralsCoordinate transformationsOrthogonality of functionsFourier seriesDifferential equationsLinear differential equationsFirst order linear DESecond order linear DEThe WronskianPower series substitutionSome special casesFrobenius' methodEulerLegendre's DEThe associated Legendre equationSolutions for Bessel's equationProperties of Bessel functionsLaguerre's equationThe associated Laguerre equationHermiteChebyshevWeberNon-linear differential equationsSturm-Liouville equationsLinear partial differential equationsGeneralSpecial casesPotential theory and Green's theoremMathematics FormularyBy ir. J.C.A. Weversc 1999, 2007 J.C.A. Wevers Version: December 30, 2007Dear reader,This document contains 66 pages with mathematical equations intended for physicists and engineers. Itis intended to be a short reference for anyone who often needs to look up mathematical equations.This document can also b e obtained from the author, Johan Wevers ([email protected]).It can also be found on the WWW on http://www.xs4all.nl/~johanw/index.html.This document is Copyright by J.C.A. Wevers. All rights reserved. Permission to use, copy and distributethis unmo dified document by any means and for any purpose except profit purposes is hereby granted.Reproducing this document by any means, included, but not limited to, printing, copying existing prints,publishing by electronic or other means, implies full agreement to the above non-profit-use clause, unlessupon explicit prior written permission of the author.The C code for the rootfinding via Newtons method and the FFT in chapter 8 are from “Numerical Recipesin C ”, 2nd Edition, ISBN 0-521-43108-5.The Mathematics Formulary is m ade with teTEX and LATEX version 2.09.If you prefer the notation in which vectors are typefaced in boldface, uncomment the redefinition of the\vec command and recompile the file.If you find any errors or have any comments, please let me know. I am always open for suggestions andpossible corrections to the mathematics formulary.Johan WeversChapter 1Basics1.1 Goniometric functionsFor the goniometric ratios for a point p on the unit circle holds:cos(φ) = xp, sin(φ) = yp, tan(φ) =ypxpsin2(x) + cos2(x) = 1 and cos−2(x) = 1 + tan2(x).cos(a ± b) = cos(a) cos(b) ∓ sin(a) sin(b) , sin(a ± b) = sin(a) cos(b) ± cos(a) sin(b)tan(a ± b) =tan(a) ± tan(b)1 ∓ tan(a) tan(b)The sum formulas are:sin(p) + sin(q) = 2 sin(12(p + q)) cos(12(p − q))sin(p) − sin(q) = 2 cos(12(p + q)) sin(12(p − q))cos(p) + cos(q) = 2 cos(12(p + q)) cos(12(p − q))cos(p) − cos(q) = −2 sin(12(p + q)) sin(12(p − q))From these equations can be derived that2 cos2(x) = 1 + cos(2x) , 2 sin2(x) = 1 −cos(2x)sin(π −x) = sin(x) , cos(π − x) = −cos(x)sin(12π − x) = c os(x) , cos(12π − x) = s in(x)Conclusions from equalities:sin(x) = sin(a) ⇒ x = a ± 2kπ or x = (π −a) ± 2kπ, k ∈ INcos(x) = cos(a) ⇒ x = a ± 2kπ or x = −a ± 2kπtan(x) = tan(a) ⇒ x = a ± kπ and x 6=π2± kπThe following relations exist be tween the inverse goniometric functions:arctan(x) = arcsinx√x2+ 1= arccos1√x2+ 1, sin(arccos(x)) =p1 − x212 Mathematics Formulary by i r. J.C.A. Wevers1.2 Hyperbolic functionsThe hyp e rbolic functions are defined by:sinh(x) =ex− e−x2, cosh(x) =ex+ e−x2, tanh(x) =sinh(x)cosh(x)From this follows that cosh2(x) − sinh2(x) = 1. Further holds:arsinh(x) = ln |x +px2+ 1| , arcosh(x) = arsinh(px2− 1)1.3 CalculusThe derivative of a function is defined as:dfdx= limh→0f(x + h) − f(x)hDerivatives obey the following algebraic rules:d(x ± y) = dx ± dy , d(xy) = xdy + ydx , dxy=ydx − xdyy2For the derivative of the inverse function finv(y), defined by finv(f(x)) = x, holds at point P = (x, f(x)):dfinv(y)dyP·df(x)dxP= 1Chain rule: if f = f(g(x)), then holdsdfdx=dfdgdgdxFurther, for the derivative s of products of functions holds:(f · g)(n)=nXk=0nkf(n−k)· g(k)For the primitive function F (x) holds: F0(x) = f(x). An overview of derivatives and primitives is:Chapter 1: Basics 3y = f(x) dy/dx = f0(x)Rf(x)dxaxnanxn−1a(n + 1)−1xn+11/x −x−2ln |x|a 0 axaxaxln(a) ax/ ln(a)exexexalog(x) (x ln(a))−1(x ln(x) − x)/ ln(a)ln(x) 1/x x ln(x) − xsin(x) cos(x) −cos(x)cos(x) −sin(x) sin(x)tan(x) cos−2(x) −ln |cos(x)|sin−1(x) −sin−2(x) cos(x) ln |tan(12x)|sinh(x) cosh(x) cosh(x)cosh(x) sinh(x) sinh(x)arcsin(x) 1/√1 − x2x arcsin(x) +√1 − x2arccos(x) −1/√1 − x2x arccos(x) −√1 − x2arctan(x) (1 + x2)−1x arctan(x) −12ln(1 + x2)(a + x2)−1/2−x(a + x2)−3/2ln |x +√a + x2|(a2− x2)−12x(a2+ x2)−212aln |(a + x)/(a − x)|The curvature ρ of a curve is given by: ρ =(1 + (y0)2)3/2|y00|The theorem of De ’l Hˆopital: if f(a) = 0 and g(a) = 0, then is limx→af(x)g(x)= limx→af0(x)g0(x)1.4 Limitslimx→0sin(x)x= 1 , limx→0ex− 1x= 1 , limx→0tan(x)x= 1 , limk→0(1 + k)1/k= e , limx→∞1 +nxx= enlimx↓0xaln(x) = 0 , limx→∞lnp(x)xa= 0 , limx→0ln(x + a)x= a , limx→∞xpax= 0 als |a| > 1.limx→0a1/x− 1= ln(a) , limx→0arcsin(x)x= 1 , limx→∞x√x = 11.5 Complex numbers and quaternions1.5.1 Complex numbersThe c omplex number z = a + bi with a and b ∈ IR. a is the real part, b the imaginary part of z.|z| =√a2+ b2. By definition holds: i2= −1. Every complex number can be written as z = |z|exp(iϕ),4 Mathematics Formulary by i r. J.C.A. Weverswith tan(ϕ) = b/a. The complex conjugate of z is defined as z = z∗:= a − bi. Further holds:(a + bi)(c + di) = (ac − bd) + i(ad + bc)(a + bi) + (c + di) = a + c + i(b + d)a + bic + di=(ac + bd) + i(bc − ad)c2+ d2Goniometric functions can be written as complex exponents:sin(x) =12i(eix− e−ix)cos(x) =12(eix+ e−ix)From this follows that cos( ix) = cosh(x) and sin(ix) = i sinh(x). Further follows from this thate±ix= cos(x) ± i sin(x), so eiz6= 0∀z. Also the theorem of De Moivre follows from this:(cos(ϕ) + i sin(ϕ))n= cos(nϕ) + i