complex-numbers - 1 tan 2 2 a--< < . Then a specific...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Chem. 540 Nancy Makri Notes on Complex Numbers A very useful representation of a complex number z is in terms of its modulus (absolute value) z and phase φ . If z x iy = + , then i z z e = where 2 2 z x y = + . This is the trigonometric form of the complex number, which is based on Euler's relation cos sin i e i α = + . The simplest way to convert a complex number to its trigonometric form is to write ( 29 2 2 cos sin z x y i = + + and equate real and imaginary parts, obtaining 2 2 2 2 cos , sin x y x y x y = = + + From this we see that the phase is an angle whose tangent is / y x , i.e., we have tan / y x = . Since the values of the trigonometric functions remain unchanged upon adding a multiple of 2 π to the angle, we write 1 tan 2 y n x - = + , n integer. Here 1 tan a - (also written as arctan a ) is the inverse (not reciprocal!) of the tangent function. To make the arc tangent function single valued, we restrict its range to the right half of the trigonometric circle, i.e.,
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 tan 2 2 a--< < . Then a specific value of n in the above expression leads to a single value for the angle. Obviously, for any real number 2 n = . The trigonometric representation facilitates many calculations. For example, powers become very easy, as ( ) k k k ik z x iy z e φ = + = . Inverses and roots are easily calculated as negative or fractional powers. For example, to find the third (cube) roots of 2 1 n i z e π = = one writes 1 3 2 /3 2 2 cos sin 3 3 n i n n z e i = = + . Distinct roots occur for n = : 1/3 1 z = 1 n = : 1/3 1 3 2 2 z i = -+ , 2 n = : 1/3 1 3 2 2 z i = --. (Other values of n produce the same results.)...
View Full Document

This note was uploaded on 02/08/2012 for the course CHEM 540 taught by Professor Mccall during the Fall '08 term at University of Illinois, Urbana Champaign.

Page1 / 2

complex-numbers - 1 tan 2 2 a--< < . Then a specific...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online