Mechanical and Electromagnetic Vibrations and Waves

APPENDICES

Appendix A

Mathematical Review

In this appendix, we designate the natural or Napierian logarithm by ln(x), the hyperbolic functions by sinh(x), cosh(x) and tanh(x). The inverse functions are designated by sinh⁻¹(x), cosh⁻¹(x), tanh⁻¹(x), sin⁻¹(x), cos⁻¹(x) and tan⁻¹(x), instead of Arcsin x, etc. The unit of angles is the radian. To simplify the notations, the partial derivatives (or derivatives) are designated by ∂_xf for ∂f/∂x, ∂²_xyf for ∂²f/∂x ∂y etc.

A.1. Expansion formulas

Taylor series near x = 0 and x = a are respectively

f(x) = f(0) + ∂_xf |_x=0 x/1! + ∂²_x f |_x=0 x²/2! + ∂³_x f |_x=0 x³/3! + ...

f(x) = f(a) + ∂_xf |_x=a (x − a)/1! + ∂²_x f |_x=a (x − a)²/2! + ∂³_x f |_x=a (x − a)³/3! + ...

Examples:

(1 + x)n = 1 + n x + n(n − 1) x2/2! + n(n − 1)(n − 2) x3/3! + …	(|x| < 1)
(1 + x)−1 = 1 − x + x2 − x3 + x4 …	(|x| < 1)
(1 + x)½ = 1 + (1/2×1!) x − (1/22×2!)x2 + (1×3/23×3!)x3…	(|x| < 1)
(x + y)n = xn + nxn−1y + n(n−1) xn−2y2/2! + n(n−1)(n−2) xn−3y3/3! + …	(|y| < |x|)

A.2. Logarithmic, exponential and hyperbolic functions

y = ex= 1+ x/1! + x2/2! + x3/3! + …,	ln(1 + x) = x −x2/2!+ x3/3! −… (x2 < 1)
sinh(x) = ½(ex−e−x) = x/1! + x3/3! + x5/5! …,	coch(x) = ½(ex+e−x) = 1+x2/2! +x4/4! …
tanh(x) = sinh(x)/coch(x) = x−x3/3 + 2x5/15…,	cosh2(x) − sinh2(x) = 1
sinh(x ± y) = sinh x cosh y ± cosh x sinh y,	cosh(x ± y) = cosh x cosh y ± sinh x sinh y
cosh(2x) = 2 cosh2x −1 = 2 sinh2x + 1,	sinh(2x) = 2 sinh x cosh x

A.3. Trigonometric functions

sin