Formulas - University of Michigan



Formulas

Trigonometric Identities:

sin(x+y) = sinx cosy + cosx siny cos(x+y) = cosx cosy - sinx siny

sinx siny = cosx cosy = sinx cosy =

sin2x = cos2x =

sinh x = (ex – e-x) cosh x = (ex + e-x)

eiy = cos y + i sin y e-iy = cos y - i sin y

sin x = (eix - e-ix) cos x = (eix + e-ix)

Cylindrical Coordinates: x = ρ cosφ y = ρ sinφ z = z

Spherical Coordinates: x = r sinθ cosφ y = r sinθ sinφ z = r cosθ

Sums and Counting

1 + x + x2 + ... + xn + ... = 1 + x + x2 + ... + xn =

1 + 2x + 3x2 + ... + n xn-1 + ... = 1 + 2x + 3x2 + ... + n xn-1 =

x + 2x2 + 3x3 + ... + n xn + ... =

# of ways to select r objects from n objects with order being taken into account =

# of ways to select r objects from n objects disregarding the order in which they are selected = ) =

Derivatives and Integrals:

ax = ax ln(a) =

tanx = sec2x

secx = tanx secx

sin-1x = ) ) dx) = sin-1x

cos-1x = ) ) dx) = - cos-1x

tan-1x = dx) = tan-1x

= uv -

= - ( + ) e-ax

= - x cos x + sin x

= x sin x + cos x

= -x2 cos x + 2x sin x + 2cos x

= x2 sin x + 2x cos x - 2sin x

= x ln(x) - x

= (a sin(bx) - b cos(bx) )

= (a cos(bx) + b sin(bx) )

Ei(x) = dt)

Ei(x) = dx) = Ei(x)

erf(x) = )

erf(x) = ) e-x2 = ,2) erf(x)

= + (Chain rule) (z = i + j(Gradient)

Integration by parts: = uv -

= g F1 - = g F1 - g' F2 + g'' F3 - + … + (-1)mg(m) Fm+1 + (-1)m+1

F1 = , F2 = , . . .

= ) exp( - )

Fourier Series:

f(x) = a0 + an cos nx for 0 < x < ( where a0 = f(x) dx an = f(x) cos nx dx

f(x) = bn sin nx for 0 < x < ( where bn = f(x) sin nx dx

f(x) = a0 + (an cos nx + bn sin nx) = An einx for - ( < x < ( where a0 = f(x) dx

an = f(x) cos nx dx = An + A-n bn = f(x) sin nx dx = i(An - A-n)

An = = f(x)e-inx dx

f(x) = a0 + an cos for 0 < x < c where a0 = f(x) dx an f(x) cos dx

f(x) = bn sin for 0 < x < c where bn = f(x) sin dx

f(x) = a0 + (an cos + bn sin) = An ein(x/c for - c < x < c where a0 = f(x) dx

an = f(x) cos dx = An + A-n bn = f(x) sin dx = i(An - A-n)

An = = f(x)e-in(x/c dx

Differential Equations:

First order linear

+ p(t)u = f(t) Multiply by e ( p(t) dt

Second order linear homogeneous

a(t) + b(t) + c(t)u = 0

u = C1u1 + C2u2 where u1 and u2 are solutions (superposition principle)

u2 = u1 dt) (second solution formula)

a + b + cu = 0 (constant coefficient)

Try u = ert ( Solve ar2 +br + c = 0 to get roots r1, r2

u = Aer1t + Ber2t if roots are real and unequal

u = (A + Bt)ert if r1 = r2 = r

u = Ae(tcos((t)+ Be(tsin((t) if roots are complex with r1 = ( + (i

ax2 + bx + cu = 0 (Euler)

Try u = xr ( Solve ar(r-1) +br + c = 0 to get roots r1, r2

u = Axr1 + Bxr2 if roots are real and unequal

u = (A + B ln(x))xr if r1 = r2 = r

u = Ax(cos(( ln(x))+ Bx(sin(( ln(x)) if roots are complex with r1 = ( + (i

Second order linear inhomogeneous

a(t) + b(t) + c(t)u = f(t)

u = up + uh up = a particular solution (superposition principle)

uh = general solution to the homogeneous equation

up = - u1 dt) + u2 dt)

Heat Equation:

ut = k(uxx + uyy + uzz) + q(x,y,z,t) ut = k(uρρ + uρ + uφφ + uzz) + q((,(,z,t)

ut = k(urr + ur + uφφ + uθθ + uθ ) + q(r,(,(,t)

Wave Equation: ytt(x,t) = a2 yxx(x,t) ztt = a2(zxx + zyy)

Boundary Value Problems - Examples

Heat Equation – One dimensional with insulated boundary conditions

= 0 < x < (, t > 0

(0, t) = 0 and ((, t) = 0 t > 0

u(x, 0) = f(x) 0 < x < (

Solution: u(x, t) = ane-n2t cos nx where an are Fourier cosine coefficients of f(x)

Wave Equation - One dimensional with zero boundary conditions and zero initial velocity

= 0 < x < (, t > 0

u(0, t) = 0 and u((, t) = 0 t > 0

u(x, 0) = f(x) 0 < x < (

(x, 0) = 0 0 < x < (

Solution: u(x, t) = bn cos nt sin nx where bn are Fourier sine coefficients of f(x)

Laplace's Equation – In a disk in polar coordinates

+ + = 0 0 < ( < 1, - ( < ( < (

u(1, () = f(() - ( < ( < (

u((, - () = u((, () and ((, - () =  ((, () 0 < x < (

u((,() "nice" as ( ( 0

Solution: u((,() = a0 + (an (n cos n( + bn (n sin n() where an and bn are Fourier coefficients of f(x)

Discrete Fourier Series

cos jx =

sin jx =

1 + cos + cos + … + cos =

sin + sin + … + sin = 0

cos cos =

sin sin =

sin cos = 0

Given f0, f1, …, fn-1, let m = . Then

fj = a0 + (ak cos + bk sin ) + am+1 (- 1)j where last term is omitted in n is odd

a0 = fj am+1 = (- 1)j fj

ak = fj cos bk = fj sin k = 1, 2, …, m

Fourier Integral Table

| |A(() = Af(() = |B(() = Bf(() = |(() = = |

|f(x) |f(x) cos (x dx |f(x) sin (x dx |f(x)e-i(x dx |

| |= (() + (- () |= i((() - (- ()) | |

| |= 2 Re[(()] if f(x) is real valued |= - 2 Im[(()] if f(x) is real valued | |

| | | | |

| | |0 | |

| | | | |

| | |0 | |

|e-x2 | e-(2/4 |0 | e-(2/4 |

| | | | |

| | | | |

| | |0 | |

|e-cx2 | e-(2/(4c) |0 | e-(2/(4c) |

| | | | |

| | | | |

| | f(x) cos (x dx | | f(x) cos (x dx |

|f(x) even | |0 |= g(() + g(- () |

| | | |where g(() = f(x)e-i(x dx |

| | | f(x) sin (x dx | f(x) sin (x dx |

|f(x) odd |0 | |= g(() - g(- () |

| | | |where g(() = f(x)e-i(x dx |

|f(- x) |Af(() |- Bf(() |(- () |

|feven(x) = |Af(() |0 |even(() |

|fodd(x) = |0 |Bf(() |odd(() |

|f(cx) (c > 0) | Af | Bf | |

|f(x - c) |(cos (c) Af(() - (sin (c) Bf(() |(sin (c) Af(() + (cos (c) Bf(() |e-i(c(() |

Bessel Functions

Jn(x) = ) =

Yn(x) = (ln() + γ) Jn(x) - ) - )

= -

where Ho = 0, H1 = 1, Hk = 1 + + + ... +

γ = = .5772.. (Euler's constant)

Jn'(x) = Jn(x) - Jn+1(x) = - Jn(x) + Jn-1(x)

= [Jo2(x) + J12(x)]

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