Electromagnetism: Maxwell Equations, Wave Propagation and Emission

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, ∂²_xyf for ∂²f/∂x ∂y, etc.

A.1. Taylor series

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

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

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

(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) = x − x2/2!+ x3/3! −... (x2 < 1)
sinh(x) = ½(ex −ex) = x/1!+x3/3!+x5/5! ...,	cosh(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 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 ...