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

Taylor series near *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)(1 + x)^{−1}= 1 −x+x^{2}−x^{3}+x^{4}…(| x| < 1)(1 + x)^{½}= 1 + (1/2×1!)x− (1/2^{2}×2!)x^{2}+ (1×3/2^{3}×3!)x^{3}…(| x| < 1)( x+y)^{n}=x^{n}+nx^{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|)

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! …, |
coch(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 |

Start Free Trial

No credit card required