complex-numbers

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

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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.,

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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.)...
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## 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.

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

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