Handbook Of Mathematics, Physics And Astronomy Data

2.9 Definite IntegralsZ x0Z 01/2 xen xx e Zπdx 2dx Z0µ1lnx10 ¶n 2.61 πx1/2dx (ex 1)2 Γ(n 1)the Gamma functionNote: for n an integer greater than 0, Γ(n 1) n!, Γ(1) 0! 12 Z u x2 e dx erf(u)π 0the Error function2 Z ax21Note that erf( ) 1, so that edx .π 0aZ 2e ax dx Z0 2n ax2x erπa1 3 5 · · · (2n 1)dx 2n 1 anZ rπ Sa(n 1, 2, 3 . . .)2x2n e ax dx 2S Z0 2x2n 1 e ax dx Z n!2an 1(a 0; n 0, 1, 2 . . .)2x2n 1 e ax dx 0 Z 0Z 0Z0x2 ln(1 e x )dx a k2π/2ndx sin(π/2n)e ax cos(kx)dx 11 x2n π 44518a2

2.10 Curvilinear Coordinate SystemsDefinitionsSpherical Coordinatesx r sin θ cos φy r sin θ sin φz r cos θCylindrical Coordinatesx r cos φy r sin φz zElements of area and volumeElements of AreaCartesian (x, y)Plane polar (r, θ)dS dx dydS r dr dθElements of VolumeCartesian (x, y, z)Spherical polar (r, θ, φ)Cylindrical polar (r, φ, z)dV dx dy dzdV r2 sin θ dr dθ dφdV r dr dφ dzmiscellaneousArea of elementary circular annulus, width dr, centred on the origin: dS 2πrdrVolume of elementary cylindrical annulus of height dz and thickness dr: dV 2πrdrdzVolume of elementary spherical shell of thickness dr, centred on the origin: dV 4πr2 dr19

ZZx r cos φy r sin φz zz direction110000 P(r, φ,z)11YYφ directiondzdrrdirectionXXCylindrical polar co-ordinatesx r sin θ cos φy r sin θ sin φz r cos φZrr dφVolume elementZdrP(r, θ,φ)θφYYr sin θ d φr dθXSpherical Polar co-ordinatesXVolume elementFigure 2.1: Coordinate Systems and Elements of volume.20

Solid angleNormal toelementary area daθrElementarysolid angle d Ωdad Ω da cos θr2steradiansFigure 2.2: Solid angle.1. The solid angle subtended by any closed surface at any point inside the surface is4π;2. The solid angle subtended by any closed surface at a point outside the surface iszero.21

2.11 Vectors and Vector AlgebraVectors are quantities with both magnitude and direction; they are combined by thetriangle rule (see Fig. 2.3).A B B A CCBθAFigure 2.3: Vector addition. orVectors may be denoted by bold type A, by putting a little arrow over the symbol A,by underlining the symbol A. Unit vectors are usually denoted by a circumflex accent(e.g. î).Magnitude etc. A (A · A) (A2x A2y A2z )The angle θ between two vectors A and B is given bycos θ A·BAx Bx Ay By Az Bz q A B (A2x A2y A2z )(Bx2 By2 Bz2 )Unit vectorsUnit vector in the direction of A A/ A Cartesian co-ordinates: î, ĵ, k̂ are unit vectors in the directions of the x, y, z cartesian axesrespectively.If Ax , Ay , Az are the cartesian components of A thenA îAx ĵAy k̂AzAddition and subtractionA B B A(Commutative law)(A B) C A (B C)22(Associative law)

BθpAFigure 2.4: Vector (or Cross) Product; the vector p is directed out of the page.ProductsScalar productA · B A B cos θ B · A(a scalar)î · î ĵ · ĵ k̂ · k̂ 1î · ĵ ĵ · k̂ k̂ · î 0A · B Ax Bx Ay By Az BzA · (B C) A · B A · CVector (or Cross) productSee Fig. 2.4A B B A ( A B sin θ)p̂ (a vector)where p̂ is a unit vector perpendicular to both A and B. Note that the vector product isnon-commutative.î ĵ k̂ĵ k̂ îk̂ î ĵî î ĵ ĵ k̂ k̂ 0Also, in cartesian co-ordinates,A B (Ay Bz Az By )î (Az Bx Ax Bz )ĵ (Ax By Ay Bx )k̂ îĵAx AyBx By k̂ Az Bz 23

Scalar triple product(A B) · C (B C) · A (C A) · B (a scalar)(Note the cyclic order: A B C) Ax Ay AzBx B y B zCx Cy Cz Ax (By Cz Bz Cy ) Ay (Bz Cx Bx Cz ) Az (Bx Cy By Cx )Vector triple productA (B C) (A · C)B (A · B)CA (B C) B (C A) C (A B) 024(a vector)(Note the cyclic order: A B C)

2.12 Complex Numbers z a ib is a complex number where a, b are real and i 1 (N.B. sometimes j isused instead of i).a is the real part of z and b is the imaginary part. Sometimes the real part of a complexquantity z is denoted by ℜ(z), the imaginary part by ℑ(z).If a1 ib1 a2 ib2 then a1 a2 and b1 b2 .Modulus and argument The modulus of z z a2 b2 .The argument of z θ arctan(b/a)Complex conjugateTo form the complex conjugate of any complex number simply replace i by i whereverit occurs in the number. Thus if z a ib then the complex conjugate is z a ib.If z Ae ix then z q A e ix . Note: z zz (a ib)(a ib) a2 b2RationalizationIf z A/B, where A and B are both complex numbers, then the quotient can be ‘rationalized’ as follows:AAB AB z BBB B 2and the denominator is now real.Polar form See Fig. 2.5. If z a ib then z a2 b2 and θ arctan(b/a). Note when evaluatingarctan(b/a), θ must be put in the correct quadrant (see Fig 2.6).eiθ cos θ i sin θsin θ eiθ e iθ2i(Euler’s identity)cos θ eiθ e iθ2z a ib z cos θ i z sin θ z exp[iθ]Ifz1 z1 exp[iθ1 ] and z2 z2 exp[iθ2 ]then z1 z2 z1 z2 exp i[θ1 θ2 ] z1 z1exp i[θ1 θ2 ] andz2 z2 25

Imaginary partof zb z #########a z cos θ#b z sin θ## θ#aReal partof zFigure 2.5: Argand diagram.If z nthen z z n z m z n z1 z2 w where w w exp[iθ] w 1/n exp[i(θ 2kπ)/n] where k 0, 1, 2 . . . (n 1) z n z m n z1 z2 DeMoivre’s theoremeinθ (cos θ i sin θ)n cos nθ i sin nθwhere n is an integerTrigonometric and hyperbolic functionssinh(iθ) i sin θcosh(iθ) cos θtanh(iθ) i tan θsin(iθ) i sinh θcos(iθ) cosh θtan(iθ) i tanh θFigure 2.6: Selecting the correct quadrant for θ arctan(b/a)26

2.13 SeriesArithmetic progression (A.P.)S a (a d) (a 2d) (a 3d) · · · (a [n 1]d)Sum over n terms isSn n[2a (n 1)d]2Geometric progression (G.P.)S a ar ar2 ar3 · · · arn 1Sum over n terms isSn If r 1 the sum to infinity isa(1 rn )(1 r)S a(1 r)Binomial theorem(a b)n an nan 1 b n(n 1) n 2 2 n(n 1)(n 2) n 3 3a b a b ···2!3!If n is a positive integer the series contains (n 1) terms. If n is a negative integer or apositive or negative fraction the series is infinite. The series converges if b/a 1.Special cases:n(n 1) 2 n(n 1)(n 2) 3x x · · · Valid for all n.2!3!1 x x2 x3 x 4 · · ·1 2x 3x2 4x3 5x4 · · ·x x2 x3 5x41 ···2816 128x 3x2 5x3 35x41 ···2816128(1 x)n 1 nx (1 x) 1 (1 x) 2 1(1 x) 2 1(1 x) 2 Maclaurin’s theorem df x2 d2 f x3 d3 f xn dn f f (x) f (0) x ··· ··· dx x 0 2! dx2 x 0 3! dx3 x 0n! dxn x 0where, for example d2 f /dx2 x 0 means the result of forming the second derivative of f (x)with respect to x and then setting x 0.27

Taylor’s theorem (x a)3 d3 f (x a)2 d2 f df ··· f (x) f (a) (x a) dx x a2!dx2 x a3!dx3 x a (x a)n dn f ··· ··· n!dxn x awhere, for example d2 f /dx2 x a again means the result of forming the second derivativeof f (x) with respect to x and then setting x a.Series expansions of trigonometric functionssin θ θ θ3 θ5 θ7 ···3!5!7!cos θ 1 θ2 θ4 θ6 ···2!4!6!For θ in radians and small (i.e. θ 1):sin θ θ 4 % for θ 30 0.52 radiansError tan θ θcos θ 1 4 % for θ 30 0.52 radiansError 4 % for θ 16 0.28 radiansError Series expansions of exponential functionse x 1 x ln(1 x) x For x small (i.e. x 1):x2 x3 x4 ···2!3!4!x2 x3 x4 ···234Convergent for all values of xConvergent for 1 x 1e x exp[ x] 1 x · · ·ln(1 x) x · · ·Series expansions of hyperbolic functions1 xx3 x5(e e x ) x ···23!5!x2 x41 x(e e x ) 1 ···cosh x 22!4!sinh x 28

L’Hôpital’s ruleIf two functions f (x) and g(x) are both zero or infinite at x a the ratio f (a)/g(a) isundefined. However the limit of f (x)/g(x) as x approaches a may exist. This may befound fromf (x)f ′ (a)lim x a g(x)g ′ (a)where f ′ (a) means the result of differentiating f (x) with respect to x and then puttingx a.Convergence TestsD’Alembert’s ratio testIn a series, Xn 1an , let the ratio R n limµ¶an 1.an If R 1 the series is convergent If R 1 the series is divergent If R 1 the test fails.The Integral TestZ A sum to infinity of an converges ifan dn is finite. This can only be applied to series1where an is positive and decreasing as n gets larger.29

2.14 Ordinary Differential EquationsGeneral points1. In general, finding a function which is ‘a solution of’ (i.e. satisfies) any particulardifferential equation is a trial and error process. It involves inductive not deductivereasoning, comparable with integration as opposed to differentiation.2. If the highest differential coefficient in the equation is the nth then the generalsolution must contain n arbitrary constants.3. The known physical conditions–the boundary conditions–may enable one particularsolution or a set of solutions to be selected from the infinite family of possiblemathematical solutions; that is boundary conditions may allow specific values to beassigned to the arbitrary constants in the general solution.4. Virtually all the ordinary differential equations met in basic physics are linear, thatis the differential coefficients occur to the first power only.Definitionsorder of a differential equationThe order of a differential equation is the order of the highest differential coefficient itcontains.degree of a differential equationThe degree of a differential equation is the power to which the highest order differentialcoefficient is raised.dependent and independent variablesOrdinary differential equations involve only two variables, one of which is referred to asthe dependent variable and the other as the independent variable. It is usually clear fromthe nature of the physical problem which is the independent and which is the dependentvariable.30

First Order Differential EquationsDirect IntegrationZdy f (x) has the solution y f (x)dx. Thus it can be solved (indxprinciple) by direct integration.The equationSeparable Variablesdy f (x)g(y), where f (x) is a function ofdxx only and g(y) is a function of y only. Dividing both sides by g(y) and integrating givesZZdy f (x) dx C, which can be used to obtain the solution of y(x).g(y)First order differential equations of the formThe linear equationdy P (x)y Q(x)dxThiscan be solved by multiplying through by an ‘integrating factor’ eI , where I RP (x)dx, so that the original equation can be rewritten asA general first order linear equation of the formd ³ I ye eI Q(x)dxSince Q and I are only functions of x we can integrate both sides to obtainIye ZQ(x)eI dxSecond Order Differential EquationsDirect IntegrationEquations of the formy d2 y f (x), can be solved by integrating twice:dx2Z ·Z f (x)dx C dx Z ·Zf (x)dx dx Cx DNote that there are two arbitrary constants, C and D.31

Homogeneous Second Order Differential Equationsadyd2 y b cy 02dxdxwhere a, b and c are constants. Letting y Aeαx , gives the auxiliary equationaα2 bα c 0This is solved for α using the quadratic equation, which gives two values for α, α1 andα2 . The general solution is the combination of the two, y Aeα1 x Beα2 x .The auxiliary equation has real roots When b2 4ac, both α1 and α2 are real. Thegeneral solution is y Aeα1 x Beα2 x .The auxiliary equation has complex roots When b2 4ac, both α1 and α2 are complex. Using Euler’s Equation, substituting C A B and D i(A B), the generalsolution can be written asy eαx (C cos(βx) D sin(βx))where α b/(2a) and β q(4ac b2 )/(2a).The auxiliary equation has equal roots When b2 4ac, there is only one α. Thegeneral solution is given by y (A Bx)eαxNon-homogeneous Second Order Differential EquationsNon-homogeneous second order differential equations are of the formadyd2 y b cy f (x)2dxdxTo solve, first solve the homogeneous equation (i.e. for right-hand side 0),d2 ydya 2 b cy 0dxdxusing the method given above to get the solutiony Aeα1 x Beα2 xwhich is known as the complementary function (CF). Then we find a particular solution(PS) for the whole equation. The general solution is CF PS.The particular solution is taken to be the same form as the function f (x).f (x) kf (x) kxf (x) kx2f (x) k sin x or k cos xf (x) ekxassumeassumeassumeassumeassume32yyyyy C Cx D Cx2 Dx E C cos x D sin x Cekx

2.15 Partial DifferentiationDefinitionIf f f (x, y) with x and y independent, thenà f x! yf (x δx, y) f (x, y)δx 0δxlim derivative with respect to x with y kept constantà f y!f (x, y δy) f (x, y)δy derivative with respect to y with x kept constant xlimδy 0The rules of partial differentiation are the same as differentiation, alwaysbearing in mind which term is varying and which are constant.Convenient notationfx 2f 2f f 2f 2f f, fxx ,f ,f ,f ,f xyyyyyx x x2 x y y y 2 y xNote that for functions with continuous derivatives fxy 2f 2f fyx x y y xTotal DerivativesTotal change in f due to infinitesimal changes in both x and y:df Ã!Ãà f x!à fdx yy!dyx!df fdy f is the total derivative of f with respect to x. dx x y y x dxÃ!Ã!df f fdx is the total derivative of f with respect to y. dy y x x y dyFor a function where each variable depends upon a third parameter, such as f (x, y) wherex and y depend on time (t):df dtà f x!yà fdx dt y33!xdydt

Maxima and Minima with two or more variablesIf f is a function of two or more variables we can still find the maximum and minimumpoints of the function. Consider a 3-d surface given by f f (x, y). We can identify thefollowing types of stationary points where gradients are zero:peak – a local maximumpit – a local minimumpass or saddle point – minimum in one direction, maximum in the 1010-10Figure 2.7: Surface plots showing a peak (left) and a saddle Point (right)At each peak, pit or pass, the function f is stationary, i.e. f f 0 x yLet f (x0 , y0 ) be a stationary point and define the second derivative test discriminant asD Ã 2f x2!Ã 2f y 2!Ã 2f x y!22 fxx fyy fxywhich is evaluated at (x0 , y0 ) and,if D 0 and fxx 0 we have a pit (minimum)if D 0 and fxx 0 we have a peak (maximum)if D 0 we have a pass (saddle point)if D 0 we do not know, have to test further comparing f (x0 , y0 ), f (x0 dx, y0 ),f (x0 , y0 dy), i.e. compare with values close to f (x0 , y0 ).34

2.16 Partial Differential EquationsThe following partial differential equations are basic to physics:Three dimensionsOne dimension 2φ x2 2φ x2 φ t1 2φc2 t2D 2 φ 0Laplace’s equation2 φ constantPoisson’s equation φ 2 φ DDiffusion equation t1 2φ2 φ 2 2Wave equationc tIn general a partial differential equation can be satisfied by a wide variety of differentfunctions, i.e. if φ f (x, t) or φ f (x, y, z), f may take many different forms which arenot equivalent ways of representing the same set of surfaces. For example, any continuous,differentiable function of (x ct) will fit the one-dimensional wave equation.‘Solving’ these partial differential equations in a particular physical context thereforeinvolves choosing not just constants but also the functions which fit the boundary conditions. Equations involving three or four independent variables, e.g. (x, y, t) or (x, y, z, t)can be solved only when the ‘boundaries’ are surfaces of some simple co-ordinate system, such as rectangular, polar, cylindrical polar, spherical polar. The partial differentialequations can then be separated into a number of ordinary differential equations in theseparate co-ordinates, and solutions can be expressed as expansions of various class

1.2 Astrophysical Quantities Symbol Quantity Value M Mass of Sun 1.989 1030 kg R Radius of Sun 6.955 108 m L Bolometric luminosity of Sun 3.846 1026 W M bol Absolute bolometric magnitude of Sun 4.75 M vis Absolute visual magnitude of Sun 4.83 T Effective temperature of Sun 5778K Spectral type of Sun G2V M J Mass of Jupiter 1.899 1027 kg R J Equatorial radius of Jupiter .