Wednesday, July 25, 2012

Math-Intro to Complex Analysis

Complex numbers are defined as points in the complex plane which consist of a real axis and an imaginary axis. By convention we denote a complex number as the point z = (x , y) where x is the real component and y is the imaginary component. z can also be written as z = x + iy.
Imaginary numbers - Imaginary numbers are just numbers who have a negative square. They are often written as a number multiplied with i, in a similar form to defining the coefficient of a unit vector. i is defined as i = (-1) 1/2 
When i is raised to some exponent it repeats a pattern of 4 different results.
= 1
= i
= -1
= -i
And then i to the 4 th will go back to one and the pattern repeats.'
Algebraic Rules-
Complex numbers follow many of the same basic properties real numbers have. They have the same commutation, association, and distribution properties that real numbers have for both addition and multiplication. i.e.
+ z = z + z 
= z 
(z + z ) + z = z + (z + z )
(z ) z = z (z )
(z + z ) = z + z 3
Complex numbers can also be treated as vectors, represented by the segment between the origin and the point. Thus add and subtract complex numbers in the same way, separately adding and subtracting the components. i.e.
1 + 2i + 3 + 4i = 4 + 5i
Again just as in vector operations, you can perform addition and subtracting using the tail to head methods, where you move the tail end of one vector without distorting it to the head end of the other vector and draw the resultant vector from the origin to the final point. However, it is more common and efficient to just use the components of the vector to calculate the resultant.
We can also define the modulus, or length of a vector in the complex plane, and once again it is done in a similar way to real vectors. The modulus of a complex number is the square of its components square rooted, denoted by |z|. i.e.
|z| = (x + y 1/2
The modulus can also be defined as the zz̅ = |z| 2.
z̅ is the complex conjugate of z. The complex conjugate is defined as if z = x + iy, z̅ = x - iy. This means the complex conjugate is the reflection of the vector defined by z over the real axis.
Exponential form-
Complex numbers can also be defined in exponential form. R and θ are used to define polar coordinates and this can be done in complex form as well.
z = r (cos (θ) + i sin (θ))
then we use the definition of Euler's formula which states:
i θ = cos (θ) + i sin (θ)
which gives us z = r e i θ 
Where r is the modulus or length of the vector that defines the complex number.
Defining complex numbers in exponential form gives grants us access to a lot of properties due to the simplicity of exponents.
Multiplying complex numbers-
1 z 2 = r 1 r 2 e i θ 1 i θ 2 
1 z 2 = r 1 r 2 e i (θ 1 + θ 2)
Thus when you multiply two complex numbers together, the resultant vector is rotated through an angle that is the sum of the two component angles, and the length is the multiplied modulus of both component vectors.
Dividing complex numbers-
1 / z 2 = (r 1 / r 2) e i θ 1 /e i θ 2 
1 z 2 = (r 1 / r 2) e i (θ 1 - θ 2)
When you divide two complex numbers, the resultant vector is rotated through an angle that is the difference of the two component angles, and the length is the divided modulus of the component vectors.
Raising to an exponent-
= r i n θ 
Raising a complex to the exponent is equivalent to raising the modulus of the vector to that power, and rotating through the angle theta of the component vector a number of times equal to the power you are raising to.
de Moivre's Formula-
This is the formula or theorem that connects complex numbers to trigonometry.
(cos(θ) + i sin(θ)) n = cos (n θ) + i sin (n θ)
This allows us to derive trigonometric identities such as the double and half angle formulas.
(cos(θ) + i sin(θ)) 2 = cos (2θ) + i sin (2θ)
cos (θ) - sin (θ) + 2 i sin(θ) cos(θ) = cos (2θ) + i sin (2θ)
Then in order to get the identities, equate the imaginary parts on both sides together and the real parts on both sides to one another.
cos (2θ) = cos (θ) - sin (θ)
i sin (2θ) = 2 i sin(θ) cos(θ)
The i's cancel in the second equation so you are left with
cos (2θ) = cos (θ) - sin (θ)
sin (2θ) = 2 sin(θ) cos(θ)
Phasors are time dependent complex vectors. They can be fully written as
A e  e iωt
Where A is the length of the vector, e  gives the initial orientation of the vector and e iωt drives the motion of the vector around in a circle in the complex plane.
This can also be written as the just A e iωt. In principle the A can also be a complex number but that is absorbed into the A e  term, we are assuming for now that the A is just a real number.
We can then separate out the real portion of this, and in doing so we get a sinusoid. This is the projection of the phasor onto the real axis.
A e iωt = A cos(ωt) + A i sin (ωt)
Re {A e iωt} = A cos(ωt)
Fourier Analysis-
F(ω) = ∫ dt V(t)e -iωt
V(t) = 1/2π ∫ dω F(ω) e iωt
These two are called Fourier transforms. The idea in the first case is to take a function of time and go to a function of frequency. The second reverses the operation and goes from a function of frequency to a function of time.
You can think of this as a form of projection, similar to the case where if we take the projection of a phasor onto the real axis we get a sinusoidal function. If you have some function in the time domain, you can approximate it with a Fourier series. A Fourier series is a series of oscillating functions with some coefficients, that when summed together form some function.
The Fourier transform when applied to a function in the time domain, gives you a function that tells you the contribution of the base waveform you are using at that frequency. This is like projecting the waveform from in the time domain into its components in the frequency domain.

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