Department of Mathematics, UMIST MATHEMATICAL FORMULA TABLES. Version 2.0 September 1999

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1 Department of Mathematics, UMIST MATHEMATICAL FORMULA TABLES Version.0 September 999

2 CONTENTS page Greek Alphabet 3 Indices and Logarithms 3 Trigonometric Identities 4 Complex Numbers 6 Hyperbolic Identities 6 Series 7 Derivatives 9 Integrals Laplace Transforms 3 Z Transforms 6 Fourier Series and Transforms 7 Numerical Formulae 9 Vector Formulae 3 Mechanics 5 Algebraic Structures 7 Statistical Distributions 9 F - Distribution 9 Normal Distribution 3 t - Distribution 3 χ (Chi-squared) - Distribution 33 Physical and Astronomical constants 34

3 GREEK ALPHABET A α alpha N ν nu B β beta Ξ ξ xi Γ γ gamma O o omicron δ delta Π π pi E ɛ, ε epsilon P ρ rho Z ζ zeta Σ σ sigma H η eta T τ tau Θ θ, ϑ theta Υ υ upsilon I ι iota Φ φ, ϕ phi K κ kappa X χ chi Λ λ lambda Ψ ψ psi M µ mu Ω ω omega INDICES AND LOGARITHMS a m a n = a m+n (a m ) n = a mn log(ab) = log A + log B log(a/b) = log A log B log(a n ) = n log A log b a = log c a log c b

4 TRIGONOMETRIC IDENTITIES tan A = sin A/ cos A sec A = / cos A cosec A = / sin A cot A = cos A/ sin A = / tan A sin A + cos A = sec A = + tan A cosec A = + cot A sin(a ± B) = sin A cos B ± cos A sin B cos(a ± B) = cos A cos B sin A sin B tan(a ± B) = tan A±tan B tan A tan B sin A = sin A cos A cos A = cos A sin A tan A = = cos A = sin A tan A tan A sin 3A = 3 sin A 4 sin 3 A cos 3A = 4 cos 3 A 3 cos A tan 3A = 3 tan A tan3 A 3 tan A sin A + sin B = sin A+B cos A B

5 sin A sin B = cos A+B sin A B cos A + cos B = cos A+B cos A B cos A cos B = sin A+B sin A B sin A cos B = sin(a + B) + sin(a B) cos A sin B = sin(a + B) sin(a B) cos A cos B = cos(a + B) + cos(a B) sin A sin B = cos(a + B) cos(a B) a sin x + b cos x = R sin(x + φ), where R = a + b and cos φ = a/r, sin φ = b/r. If t = tan x then sin x = t +t, cos x = t +t. cos x = (eix + e ix ) ; sin x = i (eix e ix ) e ix = cos x + i sin x ; e ix = cos x i sin x

6 COMPLEX NUMBERS i = Note:- j often used rather than i. Exponential Notation e iθ = cos θ + i sin θ De Moivre s theorem [r(cos θ + i sin θ)] n = r n (cos nθ + i sin nθ) n th roots of complex numbers If z = re iθ = r(cos θ + i sin θ) then z /n = n re i(θ+kπ)/n, k = 0, ±, ±,... HYPERBOLIC IDENTITIES cosh x = (e x + e x ) / sinh x = (e x e x ) / tanh x = sinh x/ cosh x sechx = / cosh x cosechx = / sinh x coth x = cosh x/ sinh x = / tanh x cosh ix = cos x sinh ix = i sin x cos ix = cosh x sin ix = i sinh x cosh A sinh A = sech A = tanh A cosech A = coth A

7 SERIES Powers of Natural Numbers n k = k= n(n + ) ; n k = n k= 6 n(n + )(n + ); k 3 = k= 4 n (n + ) n Arithmetic S n = (a + kd) = n {a + (n )d} k=0 Geometric (convergent for < r < ) S n = n k=0 Binomial (convergent for x < ) where n! (n r)!r! ( + x) n = + nx + ar k = a( rn ) r = n(n )(n )...(n r + ) r!, S = a r n! n! (n )!! x (n r)!r! xr +... Maclaurin series f(x) = f(0) + xf (0) + x! f (0) xk k! f (k) (0) + R k+ where R k+ = xk+ (k + )! f (k+) (θx), 0 < θ < Taylor series OR f(a + h) = f(a) + hf (a) + h! f (a) hk k! f (k) (a) + R k+ where R k+ = hk+ (k + )! f (k+) (a + θh), 0 < θ <. f(x) = f(x 0 ) + (x x 0 )f (x 0 ) + (x x 0) f (x 0 ) (x x 0) k f (k) (x 0 ) + R k+! k! where R k+ = (x x 0) k+ f (k+) (x 0 + (x x 0 )θ), 0 < θ < (k + )!

8 Special Power Series e x = + x + x! + x3 xr ! r! +... (all x) sin x = x x3 3! + x5 5! x7 7! ( )r x r (all x) (r + )! cos x = x! + x4 4! x6 6! ( )r x r (r)! +... (all x) tan x = x + x3 3 + x x ( x < π ) sin x = x + x x (n ) (n) x x n ( x < ) n + tan x = x x3 3 + x5 5 x7 xn ( )n +... ( x < ) 7 n + ln( + x) = x x + x3 3 x4 xn ( )n ( < x ) 4 n sinh x = x + x3 3! + x5 5! + x7 xn (all x) 7! (n + )! cosh x = + x! + x4 4! + x6 xn (all x) 6! (n)! tanh x = x x3 3 + x5 5 7x ( x < π ) sinh x = x x x x n (n ) x n ( ) +... ( x < ) n n + tanh x = x + x3 3 + x5 5 + x7 xn ( x < ) 7 n +

9 function x n e x a x (a > 0) DERIVATIVES derivative nx n e x a x lna lnx x log a x xlna sin x cos x cos x tan x cosec x sec x cot x sin x sin x sec x cosec x cot x sec x tan x cosec x x cos x tan x sinh x cosh x tanh x cosech x sech x coth x sinh x cosh x(x > ) tanh x( x < ) x + x cosh x sinh x sech x cosech x coth x sech x tanh x cosech x + x x x coth x( x > ) x

10 Product Rule d (u(x)v(x)) = u(x)dv dx dx + v(x)du dx Quotient Rule d dx ( ) u(x) = v(x) v(x) du dx u(x) dv dx [v(x)] Chain Rule d dx (f(g(x))) = f (g(x)) g (x) Leibnitz s theorem d n dx (f.g) = f (n).g+nf (n ).g () n(n ) + f (n ).g () n! n! (n r)!r! f (n r).g (r) +...+f.g (n)

11 INTEGRALS function integral f(x) dg(x) df(x) f(x)g(x) dx dx g(x)dx x n x (n = ) n+ n+ x ln x Note:- ln x + K = ln x/x 0 e x e x sin x cos x cos x sin x tan x ln sec x cosec x ln cosec x + cot x or ln tan x sec x ln sec x + tan x = ln ( ) tan π + x 4 cot x ln sin x x a + x a tan a a x a lna + x a x or x a tanh a ( x < a) x a a lnx a x + a or a coth x a ( x > a) a x sin x a (a > x ) a + x sinh x a or ln ( x + x + a ) x a sinh x cosh x cosh x a cosh x sinh x or ln x + x a ( x > a) tanh x ln cosh x cosech x ln cosechx + cothx or ln tanh x sech x tan e x coth x ln sinh x

12 Double integral f(x, y)dxdy = g(r, s)jdrds where J = (x, y) (r, s) = x r y r x s y s

13 LAPLACE TRANSFORMS f(s) = 0 e st f(t)dt function transform s t n n! s n+ e at s a ω sin ωt s + ω cos ωt sinh ωt cosh ωt t sin ωt s s + ω ω s ω s s ω ωs (s + ω ) t cos ωt H a (t) = H(t a) s ω (s + ω ) e as s δ(t) e at t n n! (s a) n+ e at sin ωt e at cos ωt e at sinh ωt e at cosh ωt ω (s a) + ω s a (s a) + ω ω (s a) ω s a (s a) ω

14 Let f(s) = L {f(t)} then L { e at f(t) } = f(s a), L {tf(t)} = d ds ( f(s)), { } f(t) L = f(x)dx if this exists. t x=s Derivatives and integrals Let y = y(t) and let ỹ = L {y(t)} then { } dy L dt { d } y L dt { t } L y(τ)dτ τ=0 = sỹ y 0, = s ỹ sy 0 y 0, = sỹ where y 0 and y 0 are the values of y and dy/dt respectively at t = 0. Time delay 0 Let g(t) = H a (t)f(t a) = f(t a) t < a t > a then L {g(t)} = e as f(s). Scale change L {f(kt)} = k f ( s k ). Periodic functions Let f(t) be of period T then L {f(t)} = e st T t=0 e st f(t)dt.

15 Convolution Let then f(t) g(t) = t x=0 f(x)g(t x)dx = t x=0 f(t x)g(x)dx L {f(t) g(t)} = f(s) g(s). RLC circuit For a simple RLC circuit with initial charge q 0 and initial current i 0, Ẽ = ( r + Ls + ) Cs ĩ Li 0 + Cs q 0. Limiting values initial value theorem final value theorem lim f(t) = lim s f(s), t 0+ s lim t 0 f(t) = lim s 0+ s f(s), f(t)dt = lim s 0+ provided these limits exist. f(s)

16 Z TRANSFORMS Z {f(t)} = f(z) = f(kt )z k k=0 function transform δ t,nt z n (n 0) e at z z e at te at T ze at (z e at ) t e at T ze at (z + e at ) (z e at ) 3 sinh at cosh at e at sin ωt e at cos ωt te at sin ωt te at cos ωt z sinh at z z cosh at + z(z cosh at ) z z cosh at + ze at sin ωt z ze at cos ωt + e at z(z e at cos ωt ) z ze at cos ωt + e at T ze at (z e at ) sin ωt (z ze at cos ωt + e at ) T ze at (z cos ωt ze at + e at cos ωt ) (z ze at cos ωt + e at ) Shift Theorem Z {f(t + nt )} = z n f(z) n k=0 zn k f(kt ) (n > 0) Initial value theorem f(0) = lim z f(z)

17 f( ) = lim z [ (z ) f(z) ] Final value theorem Inverse Formula provided f( ) exists. f(kt ) = π e ikθ f(e iθ )dθ π π FOURIER SERIES AND TRANSFORMS Fourier series f(t) = a 0 + {a n cos nωt + b n sin nωt} (period T = π/ω) n= where a n = T b n = T t0 +T t 0 t0 +T t 0 f(t) cos nωt dt f(t) sin nωt dt

18 Half range Fourier series sine series a n = 0, b n = 4 T T/ 0 f(t) sin nωt dt cosine series b n = 0, a n = 4 T T/ 0 f(t) cos nωt dt Finite Fourier transforms sine transform n= T/ f s (n) = 4 f(t) sin nωt dt T 0 f(t) = f s (n) sin nωt cosine transform f c (n) = 4 T/ f(t) cos nωt dt T 0 f(t) = f c (0) + f c (n) cos nωt n= Fourier integral Fourier integral transform ( ) lim f(t) + lim f(t) = e iωt f(u)e iωu du dω t 0 t 0 π f(ω) = F {f(t)} = π f(t) = F { f(ω) } = π e iωu f(u) du e iωt f(ω) dω

19 NUMERICAL FORMULAE Iteration Newton Raphson method for refining an approximate root x 0 of f(x) = 0 x n+ = x n f(x n) f (x n ) Particular case to find N use x n+ = ( xn + N xn ). Secant Method x n+ = x n f(x n )/ ( ) f (xn ) f (x n ) x n x n Interpolation f n = f n+ f n, δf n = f n+ f n = f n f n, µf n = f n ( fn+ + f n ) Gregory Newton Formula f p = f 0 + p f 0 + p(p ) p! f ! (p r)!r! r f 0 where p = x x 0 h Lagrange s Formula for n points y = n y i l i (x) i= where l i (x) = Πn j=,j i(x x j ) Π n j=,j =i (x i x j )

20 Numerical differentiation Derivatives at a tabular point hf 0 = µδf 0 6 µδ3 f µδ5 f 0... h f 0 = δ f 0 δ4 f δ6 f 0... hf 0 = f 0 f f f f 0... h f 0 = f 0 3 f f f Numerical Integration T rapeziumrule x0 +h x 0 f(x)dx h (f 0 + f ) + E where Composite Trapezium Rule f i = f(x 0 + ih), E = h3 f (a), x 0 < a < x 0 + h x0 +nh x 0 f(x)dx h {f 0 + f + f +...f n + f n } h (f n f 0) + h4 (f n f 0 ) where f 0 = f (x 0 ), f n = f (x 0 + nh), etc Simpson srule x0 +h x 0 f(x)dx h 3 (f 0+4f +f )+E where Composite Simpson s Rule (n even) E = h5 90 f (4) (a) x 0 < a < x 0 + h. x0 +nh x 0 f(x)dx h 3 (f 0 + 4f + f + 4f 3 + f f n + 4f n + f n ) + E where E = nh5 80 f (4) (a). x 0 < a < x 0 + nh

21 Gauss order. (Midpoint) f(x)dx = f (0) + E where E = 3 f (a). < a < Gauss order. f(x)dx = f ( 3 ) ( ) + f + E 3 where E = 35 f v (a). < a < Differential Equations To solve y = f(x, y) given initial condition y 0 at x 0, x n = x 0 + nh. Euler s forward method Euler s backward method Heun s method (Runge Kutta order ) Runge Kutta order 4. where y n+ = y n + hf(x n, y n ) n = 0,,,... y n+ = y n + hf(x n+, y n+ ) n = 0,,,... y n+ = y n + h (f(x n, y n ) + f(x n + h, y n + hf(x n, y n ))). y n+ = y n + h 6 (K + K + K 3 + K 4 ) K = f(x n, y n ) ( K = f x n + h, y n + hk ) ( K 3 = f x n + h, y n + hk ) K 4 = f(x n + h, y n + hk 3 )

22 Chebyshev Polynomials T n (x) = cos [ n(cos x) ] T o (x) = U n (x) = T n(x) n T (x) = x = sin [n(cos x)] x T m (T n (x)) = T mn (x). T n+ (x) = xt n (x) T n (x) U n+ (x) = xu n (x) U n (x) T n (x)dx = { Tn+ (x) n + T } n (x) + constant, n n f(x) = a 0T 0 (x) + a T (x)...a j T j (x) +... where a j = π π 0 f(cos θ) cos jθdθ j 0 and f(x)dx = constant +A T (x) + A T (x) +...A j T j (x) +... where A j = (a j a j+ )/j j

23 VECTOR FORMULAE Scalar product a.b = ab cos θ = a b + a b + a 3 b 3 i j k Vector product a b = ab sin θˆn = a a a 3 b b b 3 = (a b 3 a 3 b )i + (a 3 b a b 3 )j + (a b a b )k Triple products [a, b, c] = (a b).c = a.(b c) = a a a 3 b b b 3 c c c 3 a (b c) = (a.c)b (a.b)c Vector Calculus ( x, y, ) z grad φ φ, div A.A, curl A A div grad φ.( φ) φ (for scalars only) div curl A = 0 curl grad φ 0 A = grad div A curl curl A (αβ) = α β + β α div (αa) = α div A + A.( α) curl (αa) = α curl A A ( α) div (A B) = B. curl A A. curl B curl (A B) = A div B B div A + (B. )A (A. )B

24 grad (A.B) = A curl B + B curl A + (A. )B + (B. )A Integral Theorems Divergence theorem surface A.dS = volume div A dv Stokes theorem surface ( curl A).dS = contour A.dr Green s theorems volume (ψ φ φ ψ)dv = { ψ φ + ( φ)( ψ) } dv = volume surface surface ( ψ φ ) n φ ψ ds n ψ φ n ds where ds = ˆn ds Green s theorem in the plane (P dx + Qdy) = ( Q x P ) dxdy y

25 MECHANICS Kinematics Motion constant acceleration v = u + ft, s = ut + ft = (u + v)t v = u + f.s General solution of d x dt = ω x is x = a cos ωt + b sin ωt = R sin(ωt + φ) where R = a + b and cos φ = a/r, sin φ = b/r. In polar coordinates the velocity is (ṙ, r θ) = ṙe r + r θe θ and the acceleration is [ r r θ, r θ + ṙ θ ] = ( r r θ )e r + (r θ + ṙ θ)e θ. Centres of mass The following results are for uniform bodies: hemispherical shell, radius r r from centre 3 hemisphere, radius r r 8 from centre right circular cone, height h 3h 4 from vertex arc, radius r and angle θ (r sin θ)/θ from centre sector, radius r and angle θ ( r sin θ)/θ from centre 3 Moments of inertia i. The moment of inertia of a body of mass m about an axis = I + mh, where I is the moment of inertial about the parallel axis through the mass-centre and h is the distance between the axes. ii. If I and I are the moments of inertia of a lamina about two perpendicular axes through a point 0 in its plane, then its moment of inertia about the axis through 0 perpendicular to its plane is I + I.

26 iii. The following moments of inertia are for uniform bodies about the axes stated: rod, length l, through mid-point, perpendicular to rod ml hoop, radius r, through centre, perpendicular to hoop mr disc, radius r, through centre, perpendicular to disc mr sphere, radius r, diameter 5 mr Work done tb W = F. dr dt dt. t A

27 ALGEBRAIC STRUCTURES A group G is a set of elements {a, b, c,...} with a binary operation such that i. a b is in G for all a, b in G ii. a (b c) = (a b) c for all a, b, c in G iii. G contains an element e, called the identity element, such that e a = a = a e for all a in G iv. given any a in G, there exists in G an element a, called the element inverse to a, such that a a = e = a a. A commutative (or Abelian) group is one for which a b = b a for all a, b, in G. A field F is a set of elements {a, b, c,...} with two binary operations + and. such that i. F is a commutative group with respect to + with identity 0 ii. the non-zero elements of F form a commutative group with respect to. with identity iii. a.(b + c) = a.b + a.c for all a, b, c, in F. A vector space V over a field F is a set of elements {a, b, c,...} with a binary operation + such that i. they form a commutative group under +; and, for all λ, µ in F and all a, b, in V, ii. λa is defined and is in V iii. λ(a + b) = λa + λb

28 iv. (λ + µ)a = λa + µa v. (λ.µ)a = λ(µa) vi. if is an element of F such that.λ = λ for all λ in F, then a = a. An equivalence relation R between the elements {a, b, c,...} of a set C is a relation such that, for all a, b, c in C i. ara (R is reflextive) ii. arb bra (R is symmetric) iii. (arb and brc) arc (R is transitive).

29 PROBABILITY DISTRIBUTIONS Name Parameters Probability distribution / Mean Variance density function Binomial n, p P (X = r) = n! (n r)!r! pr ( p) n r, np np( p) r = 0,,,..., n Poisson λ P (X = n) = e λ λ n n!, λ λ n = 0,,,... Normal µ, σ f(x) = σ π exp{ ( x µ σ < x < ) }, µ σ Exponential λ f(x) = λe λx, λ x > 0, λ > 0 λ THE F -DISTRIBUTION The function tabulated on the next page is the inverse cumulative distribution function of Fisher s F -distribution having ν and ν degrees of freedom. It is defined by P = Γ ( ν + ν ) Γ ( ν ) Γ ( ν )ν ν ν ν x 0 u ν (ν + ν u) (ν +ν ) du. If X has an F -distribution with ν and ν degrees of freedom then P r.(x x) = P. The table lists values of x for P = 0.95, P = and P = 0.99, the upper number in each set being the value for P = 0.95.

30 ν ν : ν :

31 NORMAL DISTRIBUTION The function tabulated is the cumulative distribution function of a standard N(0, ) random variable, namely Φ(x) = π x e t dt. If X is distributed N(0, ) then Φ(x) = P r.(x x). x

32 THE t-distribution The function tabulated is the inverse cumulative distribution function of Student s t-distribution having ν degrees of freedom. It is defined by P = νπ Γ( ν + ) Γ( ν) x ( + t /ν) (ν+) dt. If X has Student s t-distribution with ν degrees of freedom then P r.(x x) = P. ν P=0.90 P=

33 THE χ (CHI-SQUARED) DISTRIBUTION The function tabulated is the inverse cumulative distribution function of a Chisquared distribution having ν degrees of freedom. It is defined by P = ν/ Γ ( ν) x 0 u ν e u du. If X has an χ distribution with ν degrees of freedom then P r.(x x) = P. For ν > 00, X is approximately normally distributed with mean ν and unit variance. ν P = P =

34 PHYSICAL AND ASTRONOMICAL CONSTANTS c Speed of light in vacuo m s e Elementary charge C m n Neutron rest mass kg m p Proton rest mass kg m e Electron rest mass kg h Planck s constant J s h Dirac s constant (= h/π) J s k Boltzmann s constant J K G Gravitational constant N m kg σ Stefan-Boltzmann constant J m K 4 s c First Radiation Constant (= πhc ) J m s c Second Radiation Constant (= hc/k) m K ε o Permittivity of free space C N m µ o Permeability of free scpae 4π 0 7 H m N A Avogadro constant mol R Gas constant 8.34 J K mol a 0 Bohr radius m µ B Bohr magneton J T α Fine structure constant (= /37.0) M Solar Mass kg R Solar radius m L Solar luminosity J s M Earth Mass kg R Mean earth radius m light year m AU Astronomical Unit m pc Parsec m year s ENERGY CONVERSION : joule (J) = electronvolts (ev)