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A Mathematical Appendix 381 A Mathematical Appendix A.1 Selected Formulae “Don’t worry about your difficulties in mathematics; I can assure you that mine are still greater.” Albert Einstein The solution of physics problems often involves mathemat- ics. In most cases nature is not so kind as to allow a precise mathematical treatment. Many times approximations are not only rather convenient but also necessary, because the gen- eral solution of specific problems can be very demanding and sometimes even impossible. In addition to these approximations, which often involve power series, where only the leading terms are relevant, ba- sic knowledge of calculus and statistics is required. In the following the most frequently used mathematical aids shall be briefly presented. 1. Power Series Binomial expansion: binomial expansion (1 ± x)m = m(m − 1) m(m − 1)(m − 2) 1 ± mx + x2 ± x3 +··· 2! 3! m(m − 1) ···(m − n + 1) + (±1)n xn +··· . n! For integer positive m this series is finite. The coefficients are binomial coefficients m(m − 1) ···(m − n + 1) m = . n! n If m is not a positive integer, the series is infinite and con- vergent for |x| < 1. This expansion often provides a simpli- fication of otherwise complicated expressions. 382 A Mathematical Appendix A few examples for most commonly used binomial ex- pansions: examples 1 1 1 5 (1 ± x)1/2 = 1 ± x − x2 ± x3 − x4 ±··· , for binomial expansions 2 8 16 128 − 1 3 5 35 (1 ± x) 1/2 = 1 ∓ x + x2 ∓ x3 + x4 ∓··· , 2 8 16 128 − (1 ± x) 1 = 1 ∓ x + x2 ∓ x3 + x4 ∓··· , (1 ± x)4 = 1 ± 4x + 6x2 ± 4x3 + x4 finite . trigonometric functions Trigonometric functions: x3 x5 x2n+1 sin x = x − + −···+(−1)n ±··· , 3! 5! (2n + 1)! x2 x4 x2n cos x = 1 − + −···+(−1)n ±··· , 2! 4! (2n)! 1 2 17 π tan x = x + x3 + x5 + x7 +··· , |x| < , 3 15 315 2 1 x x3 2x5 cot x = − − − −··· , 0 < |x| <π. x 3 45 945 exponential function Exponential function: x x2 x3 xn ex = 1 + + + +···+ +··· . 1! 2! 3! n! natural logarithm Logarithmic function: 2 3 4 n x x x + x ln(1 + x) = x − + − +···+(−1)n 1 . 2 3 4 n 2. Indefinite Integrals xn+1 powers xn x = ,(n=− ), d + 1 n 1 dx = ln x, x 1 + powers of linear functions (ax + b)n x = (ax + b)n 1 ,(n=− ), d + 1 a(n 1) dx 1 = ln(ax + b) , ax + b a A.1 Selected Formulae 383 ex dx = ex , exponential function eax x ax x = (ax − ), e d 2 1 a dx 1 eax = ln , 1 + eax a 1 + eax sin x dx =−cos x, trigonometric functions cos x dx = sin x, tan x dx =−ln cos x, ln x dx = x ln x − x. natural logarithm 3. Specific Integrals π/2 1 cos x sin x dx = , triginometric 0 2 ∞ − 2 1 π e ax dx = , Gaussian 0 2 a ∞ x dx π2 = , exponentials ex − 1 6 0 ∞ x x π2 d = x , 0 e + 1 12 ∞ ) sin ax π a> = 2 for 0 ax/x dx − π , sin 0 x 2 for a<0 1 ln x π2 dx = . logarithm 0 x − 1 6 4. Probability Distributions Binomial: binomial distribution n! f(r,n,p) = pr qn−r , r!(n − r)! r = 0, 1, 2,...,n, 0 ≤ p ≤ 1 ,q= 1 − p ; mean: r=np , variance: σ 2 = npq . 384 A Mathematical Appendix Poisson distribution Poisson: µr e−µ f(r,µ)= ,r= 0, 1, 2,... , µ>0 ; r! mean: r=µ, variance: σ 2 = µ. Gaussian distribution Gaussian: 1 (x − µ)2 f(x,µ,σ2) = √ exp − ,σ>0 ; σ 2π 2σ 2 mean: x=µ, variance: σ 2 . Landau distribution Approximation for the Landau distribution: ) * 1 1 − L(λ) = √ exp − (λ + e λ) , 2π 2 where λ is the deviation from the most probable value. 5. Errors and Error Propagation mean value Mean value of n independent measurements: n 1 x= x ; n i i=1 variance variance of n independent measurements: n n 1 1 s2 = (x −x )2 = x2 −x2 , n i i n i i=1 i=1 standard deviation where s is called the standard deviation. A best estimate for the standard estimation of the mean is s σ = √ . n − 1 If f(x,y,z) and σx , σy, σz are the function and standard independent, uncorrelated deviations of the independent, uncorrelated variables, then variables ∂f 2 ∂f 2 ∂f 2 σ 2 = σ 2 + σ 2 + σ 2 . f ∂x x ∂y y ∂z z If D(z) is the distribution function of the variable z around thetruevaluez0 with expectation value z and standard deviation σz, the quantity A.2 Mathematics for Angular Variations of the CMB 385 z+δ 1 − α = D(z) dz z−δ represents the probability that the true value z0 lies in the in- confidence interval terval ±δ around the measured value z; i.e., 100(1 − α)% measured values are within ±δ.IfD(z) is a Gaussian distri- bution one has δ 1 − α 1σ 68.27% . 2σ 95.45% 3σ 99.73% In experimental distributions frequently the full width at half full width at half maximum maximum, z, is easily measured. For Gaussian distribu- tions z is related to the standard deviation by √ z(fwhm) = 2 2ln2σz ≈ 2.355 σz . A.2 Mathematics for Angular Variations of the CMB “As physics advances farther and far- ther every day and develops new ax- ioms, it will require fresh assistance from mathematics.” Francis Bacon In this appendix the mathematics needed to describe the variations in the CMB temperature as a function of direction is reviewed. In particular, some of the important properties of the spherical harmonic functions Ylm(θ, φ) will be col- spherical harmonics lected. More information can be found in standard texts on mathematical methods of physics such as [46]. First it will be recollected what these functions are needed for in astroparticles physics. Suppose a quantity (here the temperature) as a function of direction has been dependence on the direction measured, which one can take as being specified by the standard polar coordinate angles θ and φ. This applies, e.g., for the directional measurements of the blackbody radiation. But one is not able – or at least it is highly impractical – to try to understand individually every measurement for every direction. Rather, it is preferable to parameterize the data with some function and see if one can understand the most important characteristics of this function. 386 A Mathematical Appendix When, however, a function to describe the measured temperature as a function of direction is chosen, one can- ‘periodic functions’ not take a simple polynomial in θ and φ, because this would not satisfy the obvious continuity requirements, e.g., that the function at φ = 0 matches that at φ = 2π. By using spheri- cal harmonics as the basis functions for the expansion, these requirements are automatically taken into account. Now one has to remember how the spherical harmon- important differential ics are defined. Several important differential equations of equations mathematical physics (Schrödinger, Helmholtz, Laplace) can be written in the form ∇2 + v(r) ψ = 0 , (A.1) nabla operator where ∇ is the usual nabla operator, as defined by ∂ ∂ ∂ ∇ = e + e + e . (A.2) x ∂x y ∂y z ∂z Here v(r) is an arbitrary function depending only on the ra- dial coordinate r. In separation of variables in spherical co- separation of variables ordinates, a solution of the form ψ(r,θ,φ) = R(r) Θ(θ) Φ(φ) (A.3) is tried. Substituting this back into (A.1) gives for the angu- angular parts lar parts d2Φ =−m2Φ, 2 dφ d2Θ cos θ dΘ m2 + + l(l + 1) − Θ = 0 , dθ 2 sin θ dθ sin2 θ (A.4) where l = 0, 1,...and m =−l,...,l are separation con- azimuthal solution stants. The solution for Φ is 1 Φ(φ) = √ eimφ . (A.5) 2π polar solution The solution for Θ is proportional to the associated Leg- m endre function Pl (cos θ). The product of the two angular spherical harmonic function parts is called the spherical harmonic function Ylm(θ, φ), Y (θ, φ) = Θ(θ)Φ(φ) lm 2l + 1 (l − m)! = P m(cos θ)eimφ . (A.6) 4π (l + m)! l A.2 Mathematics for Angular Variations of the CMB 387 Some of the spherical harmonics are given below: 1 Y00(θ, φ) = √ , (A.7) 4π 3 iφ Y11(θ, φ) =− sin θ e , (A.8) 8π 3 Y10(θ, φ) = cos θ, (A.9) 4π 15 2 2iφ Y22(θ, φ) = sin θ e , (A.10) 32π 15 iφ Y21(θ, φ) =− sin θ cos θ e , (A.11) 8π 5 Y (θ, φ) = (3cos2 θ − 1). (A.12) 20 16π The importance of spherical harmonics for this investi- gation is that they form a complete orthogonal set of func- tions. That is, any arbitrary function f(θ,φ) can be ex- complete orthogonal set panded in a Laplace series as of functions Laplace series ∞ l f(θ,φ)= almYlm(θ, φ) . (A.13) l=0 m=−l To determine the coefficients alm, one uses the orthogonality orthogonality relation relation ∗ = sin θ dθ dφYlm(θ, φ)Ylm (θ, φ) δl lδm m . (A.14) If both sides of (A.13) are multiplied by Ylm , integration over θ and φ leads to calculation of coefficients = ∗ alm sin θ dθ dφYlm(θ, φ)f (θ, φ) . (A.15) So, in principle, once a function f(θ,φ) is specified, the coefficients of its Laplace series can be found.
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