Appendix A

Mathematical Review

In this appendix, we designate the natural or Napierian logarithm as ln(x), and the hyperbolic functions as 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. Taylor series

Taylor series about x = 0 and x = a are, respectively,

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

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

(x + y)n = xn + n xn−1y + n(n−1)xn−2y2/2! + n(n−1)(n−2)xn−3y3/3!+ ... (|y| < |x|)

A.2. Logarithmic, exponential, hyperbolic and trigonometric functions

y = ex = 1 + x/1! + x2/2! + x3/3! +..., ln(1 + x) = xx2/2!+ x3/3! −... (x2 < 1)
sinh(x) = ½(exex) = x/1!+x3/3!+x5/5! ..., cosh(x) = ½(ex+ex) = 1+x2/2! +x4/4!
tanh(x) = sinh(x)/coch(x) = xx3/3 + 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 x = x/1! − x3/3!+ x5/5!..., cos x = 1 − x2/2! + x4/4!...
cos x = sin(π/2 − x) = −cos(π − x), sin x = cos(x –π/2) = sin(π − x)
tan ...

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