Week 1 340

Math 340


Firstly, as an introduction to complex numbers, read the wiki page.


Now, to begin our examination of the complex numbers, we start with a definition:
                                           √(-1)=i


This is the basis of complex numbers. From here, we can begin to follow all the logical paths that lead from here, and begin to explore complex numbers.


Let us first look at what happens when we raise i to integer powers:
i^2= -1
i^3=-i
i^4=1
i^n=(-1)^ (⌊n/2⌋+1)   *  i^(n mod 2)


Another way of looking at it: i^n= (i if n mod 4=1 , -1 if n mod 4=2,  -i if n mod 4=3 , else 1)


Now, ∀ z ∈ ℂ, z can be written as z=x+iy. Using simple algebra, we can see that addition and multiplication is done as follows:


let z_1= x_1 + i * y_1 
    z_2= x_2 + i * y_2


(z_1)+(z_2) = (x_1+x_2 )+ i * (y_1+y_2 )


z_1*z_2= (x_1 * x_2 - y_1 * y_2 ) + i * (y_1 * x_2 + y_2 * x_1)


But what about division?


We first define the complex conjugate: z ̅=x-iy


If we now multiply the conjugate by z, we get a real number (x^2+y^2)


This helps for division:


z_1/z_2 = (z_1 * z ̅_2) / (z_2 * z ̅_2 )=((x_1 * x_2 + y_1 * y_2 ) + i * (y_1 * x_2 - y_2 * x_1 )) / (x_2^2 + y_2^2 )




The Argand plane is now introduced:

Now, if z=x+iy, then r (being the distance from z to the origin) is √(x^2+y^2 )= √(z*z ̅ ) which is defined to be |z|.
 θ=arctan(y/x) which is defined to be the arg(z). As θ can take on an infinite amount of numbers, we define the principle argument to be θ such that -π <θ<π .


now let’s have a look at |z|.


|z_1 + z_2 | = 
                    √((z_1 * (z_1 ) ̅ ) + (z_2 * (z_2 ) ̅ ) )
                  ≤√(z_1 * (z_1 ) ̅ )+ √(z_2 * (z_2 ) ̅ )
                         =|z_1 |+|z_2 | 
 so


|z_1+z_2 | ≤ |z_1 |+|z_2 |


|z_1*z_2 |=
                  √((z_1 * (z_1 ) ̅ ) * (z_2 * (z_2 ) ̅ ) )
               = √(z_1 * (z_1 ) ̅ ) * √(z_2 * (z_2 ) ̅ )
                = |z_1 | * |z_2 |
 so


| z_1 * z_2 |=|z_1 | * |z_2 |




That covers some fairly obvious results derived from some simple definitions. We now look at complex numbers when applied to some functions:


First exponents:


Now, using a taylor expansion, we can write
e^x=1+x+x^2/2!+x^3/3!…


Now, if we apply this to complex, (z=x+iy)


e^z = e^(x+iy) 
      =e^x * e^iy
      =e^x * (1+iy+(iy)^2/2!+(iy)^3/3!…)


      =e^x * (1-y^2/2+y^4/4!-…+i(y-y^3/3!+y^5/5!…) )


Which, when considering the taylor expansion of sin and cos, is


=e^x*(cos(y)+i*sin(y) )


By then considering e^iy and e^(-iy), we can then solve simultaneously for sin and cos, getting:


(note we simply exchange y and z here, this is because these are actually definitions)


cos(z)=(e^iz+e^(-iz))/2=cosh(iz)


And


sin(z)=(e^iz-e^(-iz))/2i=isinh(iz)


We then define them as such. Note the following follows logically:


cos(iz)=cosh(iz) 
and
 sin(iz)=isinh(z)




an alternative way of writing complex numbers is polar co-ordinates, which is done as follows:


z=x+iy=r*e^iθ ,where r=|z| and θ =arg(z).


The main advantage to this becomes obvious when we look at raising z to powers and multiplying it:


Let  z_1 = r_1 * e^(i*(θ_1) )
       z_2 = r_2 * e^(i*(θ_1 ) )


 then


z_1*z_2= (r_1 * r_2 ) * e^(i((θ_1)+(θ_2)) )


And


|z_1*z_2 |=θ_1+θ_2


Now, when raising to powers:


z=r*e^iθ


z^n=(r^n) * e^(iθ*n)




and e^(θ*n)=cos(n * θ)+i * sin(n * θ)


so z^n=r^n * (cos(n * θ)+i * sin(n * θ) )


which is called the De Moivre’s theorem (although it’s really more a definition than a theorem).