# 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 ∂_{x}*f* for ∂*f*/∂*x*, ∂^{2}_{xy}*f* for ∂^{2}*f*/∂*x* ∂*y*, etc.

## A.1. Taylor series

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

*f*(*x*) = *f*(0) + ∂_{x}*f*|_{x=0}*x*/1! + ∂^{2}_{x}*f*|_{x=0}*x*^{2}/2! + ∂^{3}_{x}*f*|_{x=0}*x*^{3}/3! + ...

*f*(*x*) = *f*(*a*) + ∂_{x}*f*|_{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)*x*^{2}/2! + *n*(*n* − 1)(*n* − 2)*x*^{3}/3! + ... (|*x*| < 1)

(*x* + *y*)^{n} = *x*^{n} + *n* *x*^{n−1}*y* + *n*(*n*−1)*x*^{n−2}*y*^{2}/2! + *n*(*n*−1)(*n*−2)*x*^{n−3}*y*^{3}/3!+ ... (|*y*| < |*x*|)

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

y = e^{x} = 1 + x/1! + x^{2}/2! + x^{3}/3! +..., |
ln(1 + x) = x − x^{2}/2!+ x^{3}/3! −... (x^{2} < 1) |

sinh(x) = ½(e^{x} −e^{x}) = x/1!+x^{3}/3!+x^{5}/5! ..., |
cosh(x) = ½(e^{x}+e^{−x}) = 1+x^{2}/2! +x^{4}/4! |

tanh(x) = sinh(x)/coch(x) = x−x^{3}/3 + 2x^{5}/15..., |
cosh^{2}(x) − sinh^{2}(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 cosh^{2}x−1 = 2 sinh^{2}x+1, |
sinh(2x) = 2 sinh x cosh x |

sin x = x/1! − x3/3!+ x^{5}/5!..., |
cos x = 1 − x^{2}/2! + x^{4}/4!... |

cos x = sin(π/2 − x) = −cos(π − x), |
sin x = cos(x –π/2) = sin(π − x) |

tan ... |

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