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Mathematical formulae

전기/일반전기

by 소백산 행운지기 2011. 6. 18. 21:13

본문

Derivatives
*** General rules of differentiation
In the following, u,v,w, are functions of x; a,b,c,n are constants[restricted if indicated]; e=2.71828... is the natural base of logarithm of u[i.e. the logarithm to the base e] where it is assumed that u>0 and all angles are in radians.
  1. dc/dx = 0
  2. d(cx)/dx = c
  3. d(cxn)/dx = ncx(n-1)
  4. d(u+v+w+...)/dx = du/dx + dv/dx + dw/dx + ...
  5. d(cu)/dx = c du/dx
  6. d(uv)/dx = u dv/dx + v du/dx
  7. d(uvw)/dx = uv dw/dx + uw dv/dx + vw du/dx
  8. d(u/v)/dx = {v(du/dx)-u(dv/dx)}/v2
  9. d(un)/dx = nu(n-1) du/dx
  10. dy/dx = (dy/du) (du/dx) [chain rule]
  11. dy/dx = 1/(dx/dy)
  12. dy/dx = (dy/du)/(dx/du)
*** Derivatives of trigonometric and inverse trigonometric functions
  1. d(sin u)/dx = cos u du/dx
  2. d(cos u)/dx = -sin u du/dx
  3. d(tan u)/dx = sec2 u du/dx
  4. d(cot u)/dx = -csc2 u du/dx
  5. d(sec u)/dx = sec u tan u du/dx
  6. d(csc u)/dx = -csc u cot u du/dx
  7. d(sin-1 u)/dx = 1/√(1-u2) du/dx [-π/2 < sin-1 u < π/2]
  8. d(cos-1 u)/dx = -1/√(1-u2) du/dx [0 < cos-1 u < π]
  9. d(tan-1 u)/dx = 1/(1+u2) du/dx [-π/2 < tan-1 u < π/2]
  10. d(cot-1 u)/dx = -1/(1+u2) du/dx [0 < cot-1 u < π]
  11. d(sec-1 u)/dx = 1/{abs(u) √(u2-1)} du/dx
    = 1/{u     √(u2-1)} du/dx [if 0 < sec-1 u < π    /2]
    = -1/{u     √(u2-1)} du/dx [if π/2 < sec-1 u < π]
  12. d(csc-1 u)/dx = -1/{abs(u) √(u2-1)} du/dx
    = -1/{u     √(u2-1)} du/dx [if 0 < csc-1 u < π    /2]
    = 1/{u     √(u2-1)} du/dx [if -π/2 < csc-1 u < 0]
*** Derivatives of exponential and logarithmic functions
  1. d(log_a u)/dx = log_a (e)/u du/dx, a is not 0,1
  2. d(ln u)/dx = d(log_e u)/dx = 1/u du/dx
  3. d(au)/dx = au ln a du/dx
  4. d(eu)/dx = eu du/dx
  5. d(uv)/dx = d(ev ln u)/dx = e(v ln u) d(v ln u)/dx = vu(v-1) du/dx + uv ln u dv/dx
*** Derivatives of hyperbolic and inverse hyperbolic functions
  1. d(sinh u)/dx = cosh u du/dx
  2. d(cosh u)/dx = sinh u du/dx
  3. d(tanh u)/dx = sech2 u du/dx
  4. d(coth u)/dx = -csch2 u du/dx
  5. d(sech u)/dx = -sech u tanh u du/dx
  6. d(csch u)/dx = -csch u coth u du/dx
  7. d(sinh-1 u)/dx = 1/√(u2+1) du/dx
  8. d(cosh-1 u)/dx = 1/√(u2-1) du/dx [if cosh u > 0, u > 1]
    = -1/    √(u2-1) du/dx [if cosh u < 0, u > 1]
  9. d(tanh-1 u)/dx = 1/(1-u2) du/dx [-1 < u < 1]
  10. d(coth-1 u)/dx = 1/(1-u2) du/dx [u > 1 or u < -1]
  11. d(sech-1 u)/dx = -1/{u √(1-u2)} du/dx [if sech u > 0, 0 < u < 1]
    = 1/{u     √(1-u2)} du/dx [if sech u < 0, 0 < u < 1]
  12. d(csch-1 u)/dx = -1/{abs(u) √(1+u2)} du/dx
    = -1/{u     √    (1+u2)} du/dx [if u > 0]
    = 1/{u     √(1+u2)} du/dx [if u < 0]
*** Higher derivatives
  1. d(dy/dx)/dx = d2 y/dx2 = f"(x) = y"
  2. d(d2 y/dx2)/dx = d3 y/dx3 = f3'(x) = y3'
  3. d(d(n-1) y/dx(n-1))/dx = dn y/dxn = fn'(x) = yn'
*** Leibnitz's rule for higher derivatives of products
Let Dp stand for the operator dp/dxp so that Dp u = dp u/dxp = the pth derivative of y. Then
  • Dn(uv) = uDn v + nC1 Du D(n-1)v + nC2 D2u D(n-2)v + ... + vDn u
  • d2(uv)/dx2 = u d2 v/dx2 + 2 du/dx dv/dx + v d2 u/dx2
  • d3(uv)/dx3 = u d3 v/dx3 + 3 du/dx d2 v/dx2 + 3 d2 u/dx2 dv/dx + v d3 u/dx3
Integration
*** General rules of integration
In the following, u,v,w, are functions of x; a,b,p,q,n are constants, restricted if indicated; e=2.71828... is the natural base of logarithm of u [i.e. the logarithm to the base e] where it is assumed that u>0 [in general, to extend formulas to cases u<0 as well, replace ln u by ln(abs(u))]; all angles are in radians; all constants of integration are omitted but implies.
  1. ∫ a dx = ax
  2. ∫ af(x) dx = a ∫ f(x) dx
  3. ∫ (u+v+w+...) dx = ∫ u dx + ∫ v dx + ∫ w dx + ...
  4. ∫ u dv = uv - ∫ v du (integration by parts)
  5. ∫ f(ax) dx = 1/a ∫ f(u) du
  6. ∫ F{f(x)} = ∫ F(u) dx/du du = ∫ F(u)/f'(x) du where u=f(x)
  7. ∫ un du = u(n+1)/(n+1), n is not -1 (for n=1 see 8.)
  8. ∫ du/u = ln u if u > 0 or ln(-u) if u < 0 = ln{abs(u)}
  9. ∫ eu du = eu
  10. ∫ au du = ∫ e(u ln a) du = e(u ln a)/ln a = au/ln a, a > 0, a is not 1
  11. ∫ sin u du = -cos u
  12. ∫ cos u du = sin u
  13. ∫ tan u du = ln sec u = -ln cos u
  14. ∫ cot u du = ln sin u
  15. ∫ sec u du = ln (sec u + tan u) = ln tan(u/2 + π/4)
  16. ∫ csc u du = ln (csc u - cot u) = ln tan (u/2)
  17. ∫ sec2 u du = tan u
  18. ∫ csc2 u du = -cot u
  19. ∫ tan2 u du = tan u - u
  20. ∫ cot2 u du = -cot u - u
  21. ∫ sin2 u du = u/2 - sin (2u)/4 = 1/2 (u - sin u cos u)
  22. ∫ cos2 u du = u/2 + sin(2u)/4 = 1/2 (u + sin u cos u)
  23. ∫ sec u tan u du = sec u
  24. ∫ csc u cot u du = -csc u
  25. ∫ sinh u du = cosh u
  26. ∫ cosh u du = sinh u
  27. ∫ tanh u du = ln cosh u
  28. ∫ coth u du = ln sinh u
  29. ∫ sech u du = sin-1 (tanh u) or 2 tan-1 (eu)
  30. ∫ csch u du = ln tanh (u/2) or -coth-1 (eu)
  31. ∫ sech2 u du = tanh u
  32. ∫ csch2 u du = -coth u
  33. ∫ tanh2 u du = u - tanh u
  34. ∫ coth2 u du = u - coth u
  35. ∫ sinh2 u du = sinh (2u)/4 - u/2 = 1/2 (sinh u cosh u - u)
  36. ∫ cosh2 u du = sinh (2u)/4 + u/2 = 1/2 (sinh u cosh u + u)
  37. ∫ sech u tanh u du = -sech u
  38. ∫ csch u coth u du = -csch u
  39. ∫ du/(u2 + a2) = 1/a tan-1 (u/a)
  40. ∫ du/(u2 - a2) = 1/(2a) ln {(u-a)/(u+a)} = -1/a coth-1 (u/a), u2 > a2
  41. ∫ du/(a2 - u2) = 1/(2a) ln {(a+u)/(a-u)} = 1/a tanh-1 (u/a), u2 < a2
  42. ∫ du/√(a2 - u2) = sin-1 (u/a)
  43. ∫ du/√(u2 + a2) = ln {u + √(u2 + a2)} or sinh-1 (u/a)
  44. ∫ du/√(u2 - a2) = ln {u + √(u2 - a2)}
  45. ∫ du/{u √(u2 - a2)} = 1/a sec-1 {abs(u/a)}
  46. ∫ du/{u √(u2 + a2)} = -1/a ln {(a+√(u2+a2))/u}
  47. ∫ du/{u √(a2 - u2)} = -1/a ln {(a+√(a2-u2))/u}
  48. ∫ f(n)' g dx = f(n-1)'g-f(n-2)'g'+f(n-3)'g"- ... (-1)n ∫ fg(n)' dx
*** Important Transformation
  1. ∫ F(ax+b) dx = 1/a ∫ F(u) du, where u=ax+b
  2. ∫ F{√(ax+b)} dx = 2/a ∫ uF(u) du, where u=√(ax+b)
  3. ∫ F{(ax+b)(1/n)} dx = n/a ∫ u(n-1) F(u) du, where u=(ax+b)(1/n)
  4. ∫ F{√(a2-x2)} dx = a ∫ F(a cos u) cos u du, where x=a sin u
  5. ∫ F{√(x2+a2)} dx = a ∫ F(a sec u) sec2 u du, where x=a tan u
  6. ∫ F{√(x2-a2)} dx = a ∫ F(a tan u) sec u tan u du,where x=a sec u
  7. ∫ F(e(ax)) dx = 1/a ∫ F(u)/u du, where u=e(ax)
  8. ∫ F(ln x) dx = ∫ F(u) eu du, where u=ln x
  9. ∫ F(sin-1 (x/a)) dx = a ∫ F(u) cos u du, where u=sin-1 (x/a) similar results apply for other inverse trigonometric functions.
  10. ∫ F(sin x, cos x) dx = 2 ∫ F(2u/(1+u2),(1-u2)/(1+u2)) du/(1+u2) where u=tan(x/2)
Trigonometric functions
*** Relationships between degrees and radians
  1. 1 radian = 180 deg/π = 57.29577 95130 8232... deg
  2. 1 deg = π/180 radians = 0.01745 32925 19943 29576 92... radians
*** Relationships among trigonometric functions
  1. tan A = sin A/cos A
  2. cot A = 1/tan A
  3. sec A = 1/cos A
  4. csc A = 1/sin A
  5. sin2 A + cos2 A = 1
  6. sec2 A - tan2 A = 1
  7. csc2 A - cot2 A = 1
*** Functions of negative angels
  • sin(-A) = -sin A, same for other trigonometric functions
*** Addition Formulas
  1. sin(A+-B) = sin A cos B +- cos A sin B
  2. cos(A+-B) = cos A cos B -+ sin A sin B
  3. tan(A+-B) = (tan A +- tan B)/(1 -+ tan A tan B)
  4. cot(A+-B) = (cot A cot B -+ 1)/(cot B +- cot A)
*** Double angle formulas
  1. sin 2A = 2 sin A cos A
  2. cos 2A = cos2 A - sin2 A = 1 - 2 sin2 A = 2 cos2 A - 1
  3. tan 2A = 2 tan A / (1-tan2 A)
*** Half angle formulas
  1. sin(A/2) = √((1-cos A)/2) [if A/2 is in quadrant I or II]
    = -    √((1-cos A)/2) [if A/2 is in quadrant III or IV]
  2. cos(A/2) = √((1+cos A)/2) [if A/2 is in quadrant I or IV]
    = -    √((1+cos A)/2) [if A/2 is in quadrant II or III]
  3. tan(A/2) = √((1-cos A)/(1+cos A)) [if A/2 is in quadrant I or III]
    = -    √    ((1-cos A)/(1+cos A)) [if A/2 is in quadrant II or IV]
    = sin A/(1+cos A) = (1-cos A)/sin A = csc A - cot A
*** Multiple angle formulas
  1. sin 3A = 3 sin A - 4 sin3 A
  2. cos 3A = 4 cos3 A - 3 cos A
  3. tan 3A = (3 tan A - tan3 A)/(1-3tan2 A)
  4. sin 4A = 4 sin A cos A - 8 sin3 A cos A
  5. cos 4A = 8 cos4 A - 8 cos2 A + 1
  6. tan 4A = (4 tan A - 4 tan3 A)/(1-6tan2 A + tan4 A)
  7. sin 5A = 5 sin A - 20 sin3 A + 16 sin5 A
  8. cos 5A = 16 cos5 A - 20 cos3 A + 5 cos A
  9. tan 5A = (tan5 A -10 tan3 A + 5 tan A)/(1-10 tan2 A + 5 tan4 A)
*** Powers of trigonometric functions
  1. sin2 A = 1/2 - 1/2 cos 2A
  2. cos2 A = 1/2 + 1/2 cos 2A
  3. sin3 A = 3/4 sin A - 1/4 sin 3A
  4. cos3 A = 3/4 cos A + 1/4 cos 3A
  5. sin4 A = 3/8 - 1/2 cos 2A + 1/8 cos 4A
  6. cos4 A = 3/8 + 1/2 cos 2A + 1/8 cos 4A
  7. sin5 A = 5/8 sin A - 5/16 sin 3A + 1/16 sin 5A
  8. cos5 A = 5/8 cos A + 5/16 cos 3A + 1/16 cos 5A
*** Sum, difference and product of trigonometric functions
  1. sin A + sin B = 2 sin{1/2 (A+B)} cos{1/2 (A-B)}
  2. sin A - sin B = 2 cos{1/2 (A+B)} sin{1/2 (A-B)}
  3. cos A + cos B = 2 cos{1/2 (A+B)} cos{1/2 (A-B)}
  4. cos A - cos B = 2 sin{1/2 (A+B)} sin{1/2 (B-A)}
  5. sin A sin B = 1/2 {cos(A-B) - cos(A+B)}
  6. cos A cos B = 1/2 {cos(A-B) + cos(A+B)}
  7. sin A cos B = 1/2 {sin(A-B) + sin(A+B)}
*** General formulas
  1. sin nA = sin A {(2 cos A)(n-1) - (n-2)C1 (2cos A)(n-3) + (n-3)C2 (2cos A)(n-5) - ... }
  2. cos nA = 1/2 {(2 cos A)n - n (2 cos A)(n-2) + n/2 (n-3)C1 (2cos A)(n-4) - n/3 (n-4)C2 (2cos A)(n-6) + ... }
  3. sin(2n-1) A = (-1)(n-1)/2(2n-2) {sin (2n-1)A - (2n-1)C1 sin (2n-3)A + ...(-1)(n-1) (2n-1)C(n-1) sin A }
  4. cos(2n-1) A = 1/2(2n-2) {cos (2n-1)A + (2n-1)C1 cos (2n-3)A + ... (2n-1)C(n-1) cos A }
  5. sin(2n) A = 1/2(2n) (2n)Cn + (-1)n/2(2n-1){cos 2nA -(2n)C1 cos (2n-2)A + ...(-1)(n-1) (2n)C(n-1) cos 2A }
  6. cos(2n) A = 1/2(2n) (2n)Cn + 1/2(2n-1){cos 2nA +(2n)C1 cos (2n-2)A + ... (2n)C(n-1) cos 2A }
*** Relationships between inverse trigonometric functions
  1. sin-1 x + cos-1 x = π/2
  2. tan-1 x + cot-1 x = π/2
  3. sec-1 x + csc-1 x = π/2
  4. csc-1 x = sin-1 (1/x)
  5. sec-1 x = cos-1 (1/x)
  6. cot-1 x = tan-1 (1/x)
  7. sin-1 (-x) = -sin-1 x
  8. cos-1 (-x) = π - cos-1 x
  9. tan-1 (-x) = -tan-1 x
  10. cot-1 (-x) = π - cot-1 x
  11. sec-1 (-x) = π - sec-1 x
  12. csc-1 (-x) = -csc-1 x
*** Relationships between sides and angles of a plane triangle
  1. Laws of sines : a/sin A = b/sin B = c/sin C
  2. Laws of cosines : c2 = a2 + b2 +- 2ab cos C
  3. Laws of tangents : (a+b)/(a-b) = tan {1/2 (A+B)}/tan{1/2 (A-B)}
  4. sin A = 2/(bc) √{s(s-a)(s-b)(s-c)}, where s=1/2 (a+b+c)
*** Definition of hyperbolic functions
  1. sinh x = {ex - e(-x)} / 2
  2. cosh x = {ex + e(-x)} / 2
  3. tanh x = {ex - e(-x)} / {ex + e(-x)}
  4. coth x = {ex + e(-x)} / {ex - e(-x)}
  5. sech x = 2 / {ex + e(-x)}
  6. csch x = 2 / {ex - e(-x)}
Geometric
*** 2-Dimensional System
  1. Rectangle of length b and width a
    Area = a b; Perimeter = 2a + 2b
  2. Parallelogram of altitude h and base b (sides a,b)
    Area = b h = a b sin     θ; Perimeter = 2a + 2b
  3. Triangle of altitude h and base b (sides a,b,c)
    Area = b h / 2 = a b sin(    θ) / 2 = √{s (s-a) (s-b) (s-c)} where s = (a+b+c)/2 = semiperimeter Perimeter = a+b+c
  4. Trapezoid of altitude h and parallel sides a and (sides a,b)
    Area = h (a+b) / 2 Perimeter = a + b + h(1/sin(    θ)+1/sin(φ)) = a+b+h(csc(θ)+csc(φ))
  5. Regular polygon of n sides each of length b
    Area = n b2 cot(    π/n) / 4 = n b2 cos(π/n) / {4 sin(π/n)} Perimeter = n b
  6. Circle of radius r
    Area =     π r2; Perimeter = 2 π r
  7. Sector of circle of radius r
    Area = r2     θ / 2 [θ in radian]; Arc length s = r θ
  8. Radius of circle inscribed in a triangle of sides a,b,c
    r =     √{s (s-a) (s-b) (s-c)}/s where s = (a+b+c)/2 = semiperimeter
  9. Radius of circle circumscribing a triangle of sides a,b,c
    R = abc / [4     √{s (s-a) (s-b) (s-c)}] where s = (a+b+c)/2 = semiperimeter
  10. Regular polygon of n sides inscribed in a circle of radius r
    Area = n r2 sin(2    π/n)/2 = n r2 sin(360 deg/n)/2 Perimeter = 2nr sin(π/n) = 2nr sin(180 deg/n)
  11. Regular polygon of n sides circumscribing a circle of radius r
    Area = n r2 tan(    π/n); Perimeter = 2nr tan(π/n)
  12. Segment of circle of radius r
    Area = r2 (    θ-sin(θ))/2
  13. Ellipse of semi-major axis a and semi-minor axis b
    Area =     π a b Perimeter = 4a ∫_[from 0 to π/2] √(1-k2 sin2 (θ) d θ = 2 π √{(a2 + b2)/2} [approximately] where k=√(a2+b2)/a
  14. Segment of a parabola
    Area = 2 a b / 3 Arc length =     √(b2+16a2)/2 + b2 ln[{4a+√(b2+16a2)}/b] / 8a
*** 3-Dimensional System
  1. Rectangular parallelepiped of length a, height l, width c
    Volume = abc; Surface area = 2(ab+ac+bc)
  2. Parallelepiped of cross-sectional area A and height h (sides a,b,c)
    Volume = Ah = abc sin(    θ)
  3. Sphere of radius r
    Volume = 4     π r3 / 3; Surface area = 4 π r2
  4. Right circular cylinder of radius r and height h
    Volume =     π r2 h; Lateral surface area = 2 π r h
  5. Circular cylinder of radius r and slant height l (right height h)
    Volume =     π r2 h = π r2 l sin(θ) Lateral surface area = 2 π r l = 2 π r h/sin(θ) = 2 π r h csc(θ)
  6. Cylinder of cross-sectional area A and slant height l
    Volume = Ah = A l sin(    θ) Lateral surface area = p l = p h/sin(θ) = p h csc(θ) where p = perimeter of arbitrary shape of top or bottom
  7. Right circular cone of radius r and height h (slant height l)
    Volume =     π r2 h/3 Lateral surface area = π r2 √(r2+h2) = π r l
  8. Pyramid of base area A and height h
    Volume = A h / 3
  9. Sphere cap of radius r and height h
    Volume =     π h2 (3r-h) / 3; Surface area = 2 π r h
  10. Frustrum of right circular cone of radii a,b and height h (slant height l, Cut taper shape)
    Volume =     π h (a2 + ab + b2) / 3 Lateral surface area = π (a+b) √(h2 + (b-a)2) = π (a+b) l
  11. Spehrical triangle of angles A,B,C on sphere of radius r
    Area of triangle ABC = (A+B+C-    π) r2
  12. Torus of inner radius a and outer radius b (Donuts shape)
    Volume =     π2 (a+b) (b-a)2 / 4; Surface area = π2 (b2-a2)
  13. Ellipsoid of semi-axes a,b,c (Rugby ball shape)
    Volume = 4     π a b c / 3
  14. Paraboloid of revolution (height a, bottom radius b, Umbrella shape)
    Volume =     π b2 a / 2
Miscellaneous
*** Solution of algebraic equations
  1. Quadratic equation: ax2 + bx + c = 0
    x={-b+-    √(b2-4ac)}/2a
  2. Cubic equation: x3 + a1 x2 + a2 x + a3 = 0
    x1=S+T-a1 / 3
    x2=-(S+T)/2 - a1/3 + i     √    (3) (S-T) / 2
    x3=-(S+T)/2 - a1/3 - i     √(3) (S-T) / 2
where
Q = (3a2-a12)/9,
R = (9 a1 a2 - 27 a3 - 2 a13) / 54
S={R+√(Q3+R2)}1/3, T={R-√(Q3+R2)}1/3
D=Q3+R2
if D<0
x1 = 2 √(-Q) cos (θ/3) - a1/3
x2 = 2 √(-Q) cos (θ/3+120 deg) - a1/3
x3 = 2 √(-Q) cos (θ/3+240 deg) - a1/3
where cos(θ)=R/√(-Q3)
  1. Quartic equation: x4 + a1 x3 + a2 x2 + a3 x + a4 = 0
    Let y1 be a real root of the cubic equation
    y3 - a2 y2 + (a1 a3 - 4a4)y + (4 a2 a4 -a32 -a12 a4) = 0
    The 4 roots of
    z2 + {a1 +-     √(a12 - 4a2 + 4y1)} z/2 + {y1 -+ √(y12-4a4)}/2 = 0
*** Taylor series
  • f(x) = f(a) + f'(a)(x-a) + {f"(a)(x-a)2}/2! + ... + {f(n-1)(a)(x-a)(n-1)}/(n-1)! + Rn
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