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# 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, ∂2xyf for ∂2f/∂xy etc.

## A.1. Expansion formulas

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

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

f(x) = f(a) + ∂xf |x=a (xa)/1! + ∂2x f |x=a (xa)2/2! + ∂3x f |x=a (xa)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

 sin

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