Complex Numbers Notes

Notes for Complex Numbers

The fundamental theorem of algebra:

Gauss (1799) proved in his ‘Fundamental Theorem of Algebra’ that the field of complex numbers is algebraically closed. That is, it encloses and completes the field of numbers as a whole.

A direct result of his proof is that the following nth order polynomial:

P(z)=a_nzⁿ+a_n1zⁿ¹+…+a₁z+a₀=0,a_n0

Has n possibly-repeated complex roots for all possible complex coefficients a₀,…a_n. Equivalently, the equation as at least one complex root. The imaginary part of any root may also be zero such that the root is purely real.

Non-rigorous proof of the fundamental theorem of algebra:

The fundamental theorem of algebra can be ‘proven’ as follows:

• Generate a locus of points in the complex plane corresponding to P(z) for a given |z|=R

• Continuously deform the locus of points from large R to R=0

• Crossing the origin indicates the presence of a value |z|=R at which the value of the polynomial is zero. Thus each time the locus crosses the origin as it is deformed, the presence of a root is confirmed

Assuming that that a₀0, since a₀=0 gives a trivial root z=0 in which case P(z) is simply divided by z to form a polynomial of order n1:

• For large R, the locus of P(z) is almost a circle centred on the origin, which wraps around the origin n times

• At very small R, the locus of P(z) is almost a single circle of radius a₁z centred on a₀

• Thus, during the continual deformation of one to the other, the locus must cross the origin n times

• Hence P(z) must have n complex roots

The imaginary exponential:

The most important result in complex number analysis, which underpins the algebraic rules used to manipulate such numbers, concerns the exponential of an imaginary number:

e^iθ

The value of this exponential is derived using the Taylor Series for e^x:

$$e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \frac{x^{5}}{5!} + \ldots$$

$$\therefore e^{i\theta} = 1 + i\theta - \frac{\theta^{2}}{2!} - \frac{i\theta^{3}}{3!} + \frac{\theta^{4}}{4!} + \frac{i\theta^{5}}{5!} + \ldots$$

$$\therefore e^{i\theta} = \left( 1 - \frac{\theta^{2}}{2!} + \frac{\theta^{4}}{4!} + \ldots \right) + i\left( \theta - \frac{\theta^{3}}{3!} + \frac{\theta^{5}}{5!} + \ldots \right)$$

e^iθ=cosθ+isinθ

$$\boxed{\mathbf{\therefore r}\mathbf{e}^{\mathbf{i\theta}}\mathbf{= r}\left( \mathbf{\cos}\mathbf{\theta}\mathbf{+ i}\mathbf{\sin}\mathbf{\theta} \right)}$$

Algebraic laws derived from the imaginary exponential:

The algebraic laws which allow for the manipulation of complex numbers, including de Moivre’s Theorem, are derived from the definition of the imaginary exponential above:

Let z₁=r₁e^iθ₁=r₁(cosθ₁+isinθ₁)

Let z₂=r₂e^iθ₂=r₂(cosθ₂+isinθ₂)

Multiplication:

z₁z₂=(r₁e^iθ₁)(r₂e^iθ₂)=r₁r₂e^{i(θ₁+θ₂)}

z₁z₂=r₁r₂(cos(θ₁+θ₂)+isin(θ₁+θ₂))

r₁cisθ₁r₂cisθ₂=r₁r₂cis(θ₁+θ₂)

Division:

$$\frac{z_{1}}{z_{2}} = \frac{r_{1}e^{i\theta_{1}}}{r_{2}e^{i\theta_{2}}} = \frac{r_{1}}{r_{2}}e^{i\left( \theta_{1} - \theta_{2} \right)}$$

$$\therefore\frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}}(\cos\left( \theta_{1} - \theta_{2} \right) + i\sin\left( \theta_{1} - \theta_{2} \right))$$

$$\mathbf{\therefore}\frac{\mathbf{r}_{\mathbf{1}}\mathbf{cis}\mathbf{\theta}_{\mathbf{1}}}{\mathbf{r}_{\mathbf{2}}\mathbf{cis}\mathbf{\theta}_{\mathbf{2}}}\mathbf{=}\frac{\mathbf{r}_{\mathbf{1}}}{\mathbf{r}_{\mathbf{2}}}\mathbf{cis}{\mathbf{(\theta}_{\mathbf{1}}\mathbf{-}\mathbf{\theta}_{\mathbf{2}}\mathbf{)}}$$

De Moivre’s Theorem:

(re^iθ)ⁿ=rⁿe^niθ

[r(cosθ+isinθ)]ⁿ=rⁿ(cos(nθ)+isin(nθ))

(rcisθ)ⁿ=rⁿcis(nθ)

Manipulating numbers in the complex plane:

The manipulation of complex numbers may be represented by geometric operations in the complex plane:

Addition/subtraction:

Addition or subtraction of z₁=a+ib to or from z₂=c+id to give z₃:

• Corresponds to a shift in co-ordinates of (a,b) in the Argand diagram

• Geometrically equivalent to the addition/subtraction of position vectors z₁ and z₂ from the origin of the Argand diagram.

Multiplication:

Multiplication of z₁=r₁e^iθ₁ with z₂=r₂e^iθ₂ to give z₃:

• Corresponds to a rotation by angle θ₂ about the origin, coupled with a change in distance from the origin by factor r₂:

z₁z₂=r₁r₂e^iθ₁e^iθ₂=r₁r₂e^{i(θ₁+θ₂)}

• If |z₂|=1, the complex number is simply rotated about the origin by angle θ₂

Curves in the Argand Plane:

Since the real and imaginary axes of the Argand plane are geometrically equivalent to the x and y axes of a Cartesian plane, the locus of points |z+a+ib|=r in the Argand plane are equivalent to the circle in the Cartesian plane given by the equation (x+a)²+(y+b)²=r²:

|z+a+ib|=r,z=x+yi

|x+a+i(y+b)|=r

(x+a)²+(y+b)²=r²

Transformations of circles in the Argand Plane:

Translation:

Dilation:

Rotation:

Logarithms of complex numbers:

The imaginary exponential allows the domain of logarithms to extend to any number, not just positive, real numbers. The logarithm of any number re^iθ can be derived as follows:

w=lnz=ln(re^iθ)=lnr+iθ

Or, more generally:

$$\boxed{\mathbf{\ln}{\mathbf{(r}\mathbf{e}^{\mathbf{i\theta}}\mathbf{)}}\mathbf{=}\mathbf{\ln}\mathbf{r}\mathbf{+ i(\theta + 2}\mathbf{\pi n)}}$$

For example, we can find the logarithm of -2 to base 10:

$$\log_{10}{- 2} = \frac{\ln{- 2}}{\ln 10} = \frac{\ln{2e^{i\pi}}}{\ln 10}$$

$$\therefore\log_{10}{- 2} = \frac{\ln 2 + \pi i}{\ln 10}$$

General powers:

A ‘general power’ is a power expression z₁^z₂ formed from two complex numbers z₁ and z₂.

In order to evaluate such a general power, the natural logarithm of a complex number, lnz, is used, by noting that:

z₁=e^lnz₁

z₁^z₂=(e^z₂lnz₁)

For example, complex algebra can be used to manipulate general powers as follows:

Find all solutions of z=i¹ⁱ:

$$z = i^{1 - i}\therefore z = \left( e^{\ln i} \right)^{1 - i} = e^{(1 - i)\ln e^{\left( \frac{\pi}{2} + 2k\pi \right)i}},\ \ k = 0, \pm 1, \pm 2\ldots$$

$$\therefore z = e^{(1 - i)i(\frac{\pi}{2} + 2k\pi)} = e^{i\left( \frac{\pi}{2} + 2k\pi \right)}e^{\frac{\pi}{2} + 2k\pi},\ \ k = 0, \pm 1, \pm 2\ldots$$

$$\therefore z = e^{\frac{\pi}{2} + 2k\pi}\left( \cos\left(...

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