O'Reilly logo

Electromagnetism: Maxwell Equations, Wave Propagation and Emission by Tamer Bécherrawy

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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 ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required