Tab-integrais

Mathematics 1. If ≠ ′, no solution, if = ′ = , unique solution if = ′ < , many solutions. (non-homogeneous) 2. If = , trivial solution, if < ,then (−) linearly independent solutions. (Many solutions) and if < , then many solutions. 3. (+ ℎ) = () + ℎ ′ () + ℎ 2 2! ′′ () + ℎ 3 3! ′′′ () + … … … ∞ 4. If − 2 > 0 and < 0 (,) have maximum, if − 2 > 0 and > 0 (,) have minimum at(,) and if − 2 < 0, then saddle point. If − 2 = 0, ℎ investigation is required to decide. 5. ∫ (∅ +) = ∫ ∫ (− ∅) (Green's) 6. ∫. ℝ=∫. (Stokes) 7. ∫. = ∫ (Gauss) 8. ∫ = ∫ ∫ + 9. If + = 0 be a homogeneous equation in and , then 1 + an integrating factor 10. If the equation of the type 1 () + 2 () = 0. If the equation + = 0 be of this type then 1 − is an integrating factor 11. If − be a function of x only = () say then ∫ () is an integrating factor 12. If − be a function of y only = () say then ∫ () is an integrating factor. 13. ∫ = + ∫ terms of N not containing x dy = c 14.. . = 1 () = 1 () , () ≠ 0, if () = 0, ℎ. . = 1 ′ () , ′ () ≠ 0 15.. = 1 (2) sin(+) = 1 (− 2) , (− 2) ≠ 0, if (− 2) = 0, then. . = 1 ′ (− 2) sin(+), ′ (− 2) ≠ 0 16.. . = 1 () = 1 (+) 17.. = 1 () = [ ()] −1 , 18. (1 +) −1 = 1 − + 2 − ⋯ 19. (1 −) −1 = 1 + + 2 + ⋯ 20. + 1 −1 −1 −1 + ⋯ −1 + = , = , = , 2 2 2 = (− 1) , 3 3 3 = D(D − 1)(D − 2) 21. (∫ ℎ(,) () ()) = ∫ ℎ(,) () () + ℎ[ (), ] − ℎ[ (), ] 22. { ()} = ∫ − ∞ 0 () 23. (1) = 1 24. () = ! +1 25. () = 1 − 26. (sin) = 2 + 2 27. (cos) = 2 + 28. (sinh) = 2 − 2 29. (cosh) = 2 − 2 30. { ()} = ̅ (−) 31. (+) = () then { ()} = ∫ − () 0 1− − 32. { ′ ()} = ̅ () − (0) 33. { ()} = ̅ () − −1 (0) − −2 ′ (0) − ⋯ … …. . −1 (0) 34. {∫ () 0 } = 1 ̅ () 35. { ()} = (−1). [ ̅ (s)] 36. { 1 ()} = ∫ ̅ (s) ∞ 37. () = 0 2 + ∑ ∞ =1 cos + ∑ ∞ =1 sin 38. 0 = 1 ∫ () +2 , = 1 ∫ () cos +2 , = 1 ∫ () sin +2 39. () = 0 2 + ∑ ∞ =1 cos + ∑ ∞ =1 sin 40. 0 = 1 ∫ () +2 , = 1 ∫ () cos +2 , = 1 ∫ () sin +2 41. () = ∑ ∞ =1 sin , where = 2 ∫ () sin 0 42. () = 0 2 + ∑ ∞ =1 cos where, 0 = 2 ∫ () 0 , = 2 ∫ () cos 0 43. = ∑ () = ∫ () ∞ −∞ 44. 2 = ∑ (−) 2 () 2 = ∫ (−) 2 () ∞ −∞ 45. : = : 2 = (Poisson's distribution) 46. () = 1 √2 − 1 2 (−) 2 (Normal distribution) 47. = + , ∑ = + ∑ , ∑ = ∑ + ∑ 2 48. 50. () 0 = 1 ℎ [∆ 0 − 1 2 ∆ 2 0 + 1 3 ∆ 3 0 − 1 4 ∆ 4 0 + ⋯ ] 51. () = 1 ℎ [∇ + 1 2 ∇ 2 + 1 3 ∇ 3 + 1 4 ∇ 4 + ⋯ ] 52. +1 = − () ′ () (Newton-Raphson) 53. ∫ () 0 + ℎ 0 = ℎ 2 [ 0 + + 2(1 + 2 + ⋯. . + −1)] (Trapezoidal) 54. ∫ () 0 + ℎ 0 = ℎ 3 [(0 +) + 4(1 + 3 + ⋯ −1) + 2(2 + 4 + ⋯ −2)] (Simpson's) 55. = − − 12 ℎ 2 ′′ () = (ℎ 2) (Trapezoidal) 56. = − − 180 ℎ 4 () = (ℎ 4) (Simpson's) 57. +1 = + ℎ. (,) where = (,) (Euler's) 2

Mathematics of Computation, 1968

1. Second order; linear 2. Third order; nonlinear because of (dy/dx) 4 3. Fourth order; linear 4. Second order; nonlinear because of cos(r + u) 5. Second order; nonlinear because of (dy/dx) 2 or 1 + (dy/dx) 2 6. Second order; nonlinear because of R 2 7. Third order; linear 8. Second order; nonlinear because of ˙ x 2 9. Writing the differential equation in the form x(dy/dx) + y 2 = 1, we see that it is nonlinear in y because of y 2. However, writing it in the form (y 2 − 1)(dx/dy) + x = 0, we see that it is linear in x. 10. Writing the differential equation in the form u(dv/du) + (1 + u)v = ue u we see that it is linear in v. However, writing it in the form (v + uv − ue u)(du/dv) + u = 0, we see that it is nonlinear in u. 11. From y = e −x/2 we obtain y = − 1 2 e −x/2. Then 2y + y = −e −x/2 + e −x/2 = 0. 12. From y = 6 5 − 6 5 e −20t we obtain dy/dt = 24e −20t , so that dy dt + 20y = 24e −20t + 20 6 5 − 6 5 e −20t = 24. 13. From y = e 3x cos 2x we obtain y = 3e 3x cos 2x − 2e 3x sin 2x and y = 5e 3x cos 2x − 12e 3x sin 2x, so that y − 6y + 13y = 0. 14. From y = − cos x ln(sec x + tan x) we obtain y = −1 + sin x ln(sec x + tan x) and y = tan x + cos x ln(sec x + tan x). Then y + y = tan x. 15. The domain of the function, found by solving x + 2 ≥ 0, is [−2, ∞). From y = 1 + 2(x + 2) −1/2 we have (y − x)y = (y − x)[1 + (2(x + 2) −1/2 ] = y − x + 2(y − x)(x + 2) −1/2 = y − x + 2[x + 4(x + 2) 1/2 − x](x + 2) −1/2 = y − x + 8(x + 2) 1/2 (x + 2) −1/2 = y − x + 8.

American Journal of Physics, 1988

This paper attempts to show how very subtle Mayan compositions were. The use of Form, Imagery, and Gesture signs to create the structural organization of the composition can be almost subliminal. The imagery can distract from the composition's actual meaning at least for present day researchers.