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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