Solving Differential Equations Involving Complex Numbers

You can only grow if you’re willing to feel awkward and uncomfortable when you try something new, Brian Tracy

Recall

Definition. A differential equation is an equation that involves one or more dependent variables, their derivatives with respect to one or more independent variables, and the independent variables themselves, e.g., $\frac{dy}{dx} = 3x +5y, y’ + y = 4xcos(2x), \frac{dy}{dx} = x^2y+y, etc.$

It involves (e.g., $\frac{dy}{dx} = 3x +5y$):

• Dependent variables: Variables that depend on one or more other variables (y).
• Independent variables: Variables upon which the dependent variables depend (x).
• Derivatives: Rates at which the dependent variables change with respect to the independent variables, $\frac{dy}{dx}$

First-Order Linear Ordinary Differential Equations (ODEs)

Definition. A first-order linear ordinary differential equation is an ordinary differential equation (ODE) involving an unknown function y(x), its first derivative y′, and functions of the independent variable x, which can be written in the general form:: a(x)y' + b(x)y = c(x) where:

• y = y(x) is the unknown function of the independent variable x.
• y′ = $\frac{dy}{dx}$ is the derivative of y with respect to x.
• a(x), b(x), and c(x) are known functions of x.

These equations are termed “linear” because the unknown function y and its derivative y’ appear to the first power and are not multiplied together or composed in any nonlinear way.

If the function c(x)=0 for all x in the interval of interest, the equation simplifies to: a(x)y’ + b(x)y = 0. Such an equation is called a homogeneous linear differential equation.

The Existence and Uniqueness Theorem provides crucial insight into the behavior of solutions to first-order differential equations ODEs. It states that if:

• The function f(x, y) (the right-hand side of the ODE) in the equation y’ = f(x, y) is continuous in some neighborhood around a point (x₀, y₀) and
• Its partial derivative with respect to y, denoted as $\frac{∂f}{∂y}$, is also continuous near (x₀, y₀).

Then, the differential equation y' = f(x, y) has a unique solution to the initial value problem through the point (x₀, y₀), meaning that it satisfies the initial condition y(x₀) = y₀.

This theorem ensures that under these conditions, the solution exists and is unique near x = x₀.

Complex Numbers

Complex numbers are fundamental in various fields of mathematics, physics, and engineering.

A complex number is any number that can be expressed or written in the form z = a + bi, where a and b are real numbers, and i is the imaginary unit, defined by the property i² = −1.

In the complex number z = a + bi: a is called the real part of the complex number, denoted as Re(z) = a. b is called the imaginary part, denoted as Im(z) = b.

Example: z = 3 +4i, the real part is Re(z) = 3 and the imaginary part is Im(z) = 4.

The complex conjugate of a complex number z = a + bi is denoted by $\bar z$ and is defined as $\bar z = a -bi$.

Multiplying a complex number by its conjugate yields a real number: $z·\bar z = (a + bi)(a -bi) = a^2 + b^2$, e.g., z = 3 + 4i, $z·\bar z = (3 + 4i)(3 - 4i) = 3^2 + 4^2 = 9+16 = 25.$

Addition, Subtraction, and Multiplication

• Addition: (a +bi) + (c +di) = (a +c) + (b +d)i
• Subtraction: (a +bi) − (c +di) = (a −c) + (b −d)i
• Multiplication. Multiply complex numbers by applying the distributive property and using i² = −1: (a + bi)(c + di) = (ac −bd) + (ad + bc)i

Division of Complex Numbers

To divide complex numbers, we eliminate the imaginary unit from the denominator by multiplying both the numerator and denominator by the complex conjugate of the denominator.

Example: Calculate $\frac{2+i}{1-3i}$

$\frac{2+i}{1-3i}$ =[To simplify the expression, multiply both numerator and denominator by the conjugate of the denominator:] $\frac{2+i}{1-3i}·\frac{1+3i}{1+3i} = \frac{-1+7i}{10} =[\text{Thus, the simplified form is:}] \frac{-1}{10}+\frac{7}{10}i$.

Polar representation of Complex Numbers

A complex number z = a +bi can also be expressed in polar form, which uses its modulus (magnitude) r and argument (angle) θ:

• The modulus r = |z| = $\sqrt{a^2+b^2}$. It represents the distance of the complex number from the origin in the complex plane.
• The argument $θ = arg(z) = tan^{-1}(\frac{b}{a})$. It is the angle between the positive real axis and the line representing the complex number. The angle θ should be determined in the correct quadrant, considering the signs of a and b.

The polar form of a complex number is: z = a + bi = rcos(θ) + rsin(θ)i = r(cos(θ) + isin(θ)) =[This can be written compactly using Euler’s formula] re^iθ

Euler’s formula is a fundamental formula in complex analysis. It establishes the mathematical relationship between trigonometric functions and the complex exponential function. It states that e^iθ = cos(θ) + isin(θ). This allows us to represent complex numbers compactly in exponential form. It is essential in solving differential equations with trigonometric inputs.

Laws of Exponents for Complex Numbers

The laws of exponents apply to complex numbers in exponential form, similar to real numbers.

1. Product Rule: a^m·aⁿ = a^m+n.
2. Quotient Rule: a^m÷aⁿ = a^m-n
3. Power of a Power Rule: (a^m)ⁿ = a^mn
4. Power of a Product Rule: (ab)^m = a^mb^m
5. Power of a Quotient Rule: (a/b)^m = (a^m)/(b^m)
6. Zero Exponent Rule: a⁰ = 1 for a ≠ 0
7. Negative Exponent Rule: a^-m = 1/(a^m)
8. Fractional Exponent Rule: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$

Multiplication and Division of Complex Numbers in Polar Form

Multiplying complex numbers in polar form is simplified using exponentials.

Given two complex numbers in polar form: $z_1 = r_1e^{iθ_1},~ z_2 = r_2e^{iθ_2}$. The product is: $z_1·z_2 = r_1r_2e^{iθ_1}·e^{iθ_2}=[\text{Laws of Exponents. Product Rule.}]r_1r_2e^{iθ_1+iθ_2}=r_1r_2e^{i(θ_1+θ_2)}$. This shows that:

• The modulus of the product is the product of the moduli: |z₁z₂| = r₁r₂.
• The argument of the product is the sum of the arguments, arg(z₁z₂) = θ₁ + θ₂

Division: $\frac{z_1}{z_2} = \frac{r_1e^{iθ_1}}{r_2e^{iθ_2}} =[\text{Laws of Exponents. Product Rule.}] \frac{r_1}{r_2}e^{iθ_1-iθ_2}=\frac{r_1}{r_2}e^{i(θ_1-iθ_2)}$. In words, moduli divide and arguments subtract.

Differentiation and Integration of Complex Functions

Differentiation of Complex Functions A complex-valued function can be expressed in terms of its real and imaginary parts: y(t) = u(t) + iv(t), where:

• u(t) is the real part, the real-valued function of t.
• v(t) is the imaginary part, the real-valued function of t.
• i is the imaginary unit, with i² = -1.

Differentiation Rule. The derivative of y(t) with respect to t is obtained by differentiating the real and imaginary parts separately: $\frac{d}{dt}y(t) = \frac{d}{dt}[u(t)+iv(t)] = \frac{du}{dt}+i\frac{dv}{dt}$ ↭ D(u + iv)= Du + iDv. This rule follows from the linearity of differentiation.

Example: Differentiating the Complex Exponential Function. Consider the function y(t) = e^it. $\frac{d}{dt}e^{it} =[\text{Using Euler’s formula}] \frac{d}{dt}(cos(t) + isin(t)) =[\text{Differentiation Rule}] \frac{d}{dt}(cos(t)) + i\frac{d}{dt}(sin(t)) = -sin(t) + icos(t) = i(cos(t) + isin(t)) =[\text{Using Euler’s formula}] ie^{it}$, which is consistent with the property of exponential functions.

$\frac{d}{dt}e^{it} = ie^{it}$. The derivate of e^it is ie^it. Multiplying by i rotates the complex number e^it by 90° in the complex plane. This property is fundamental in solving differential equations involving complex exponentials.

Integration follows similar principles, integrating the real and imaginary parts separately.

Consider $\int e^{it}dt = \int (cos(t)+isin(t))dt = sin(t)-icos(t)+C = -i(cos(t)+sin(t)i)+C =-ie^{it}+C$

Solving Differential Equations Involving Complex Numbers

Consider the differential equation: $\frac{dy}{dt} = iy$ where y is a function of t, and i is the imaginary unit with the initial condition y(0) = 1.

General Solution.

This is a first-order linear ordinary differential equation (ODE). We can solve it using the method of separation of variables.

$\frac{dy}{dt} = iy ↭[\text{Separate variables}] \frac{dy}{y} = idt ↭[\text{Integrate both sides}] \int \frac{dy}{y} = \int idt ↭ lny = it + C$ where C is the constant of integration. Solve for y: $y = e^{it+C} = e^Ce^{it}$. Let K = e^C, the general solution is y(t) = Ke^it.

Applying the initial condition. Given y(0) = 1 ↭ y(0) = Ke^i·0 = K·1 = K = 1. Final solution: y(t) = e^it

Verification: $\frac{d}{dt}y(t) = \frac{d}{dt}e^{it} = ie^{it} = i(y(t)), y(0)=e^{i·0} = cos(0) + isin(0) = 1 + i·0 = 1$. This confirms that y(t) = e^it satisfies both the differential equation and the initial condition.

Exponential Expressions with Complex Exponents

Understanding complex exponents is crucial for working with complex functions.

General Expression. Consider the general exponential expression e^a+ib where a and b are real numbers, a, b ∈ ℝ.

e^a+ib =[Using properties of exponents, this expression can be rewritten as a product of two simpler exponentials] e^a·e^ib where e^a is a real number representing the magnitude and e^ib is a complex exponential, representing rotation in the complex plane.

Euler’s formula relates complex exponentials to trigonometric functions: e^ib = cos(b) + isin(b).

Combining the expressions, the complex exponential e^a+ib can be written as: e^a+ib = e^a(cos(b) + isin(b)).

r = e^a is the magnitude or modulus, and θ = b is the angle.

It representsa point in the complex plane at a distance e^a from the origin and at an angle b from the positive real axis.

Polar Representation of Complex Numbers

Complex numbers can be represented in polar form, which is particularly useful for multiplication and division.

Given a complex number α = e^a+ib = e^a(cos(b)+isin(b)). It can also be represented in polar form as re^iθ = r(cos(θ) + isin(θ)) (Refer to Figure iii for a visual representation and aid in understanding it), where:

• r = |α| = e^a is the modulus or magnitude of the complex number. It is the distance from the origin to the point α in the complex plane.
• θ = arg(α) = $tan^{-1}(\frac{sin(b)}{cos(b)})=tan^{-1}(tan(b))$ = b is the argument or the angle between the positive real axis and the line representing α.

Integration of Real Functions Using Complex Exponentials

Complex exponentials can simplify the integration of real functions involving trigonometric terms.

Consider the integral: $\int e^{-x}cos(x)dx$. We can use complex exponentials to simplify the integral by expressing cos(x) as the real part of e^ix, cos(x) = $Re(e^{ix})$

$\int e^{-x}cos(x)dx$ =[Express the integral in terms of e^ix] $Real(\int e^{-x}e^{ix}dx) = Real(\int e^{(-1+i)x}dx)$

Integrate the complex exponential: $\int e^{(-1+i)x}dx = \frac{e^{(-1+i)x}}{-1+i} + C$ where C is the constant of integration.

Simplifying the expression: $\frac{e^{(-1+i)x}}{-1+i} = \frac{1}{-1+i}e^{-x}e^{ix} = \frac{1}{-1+i}e^{-x}(cos(x)+isin(x))=\frac{1}{-1+i}\frac{-1-i}{-1-i}e^{-x}(cos(x)+isin(x)) = \frac{-1-i}{2}e^{-x}(cos(x)+isin(x))$ 🚀

$\int e^{-x}cos(x)dx = Real (\frac{e^{(-1+i)x}}{-1+i}) =[\text{Taking the real part 🚀}] \frac{e^{-x}}{2}(-cos(x)+sin(x))+C$

N-th root of unity

The n-th roots of unity are the complex solution to the equation: xⁿ = 1. This equation asks for all complex numbers z such that raising z to the power n yields 1. This can be approached using Euler’s formula, which connects complex exponentials with trigonometric functions.

In the real number system, the only solution to this equation is x = 1, since 1 is the only real number that, when raised to any power, results in 1.

However, in the complex domain, there are n distinct solutions to this equation, distributed symmetrically on the unit circle (the set of points in the complex plane at a distance of 1 from the origin). These roots are known as the n-th roots of unity.

Finding the N-th Roots of Unity

The number 1 can be represented using Euler’s formula by choosing θ = 0: 1 = $e^{i·0} = cos(0)+ isin(0) = 1 + i·0$. The complex equation zⁿ = 1 can be rewritten using exponential form: $z^n = e^{i·0} = e^{i·2πk}$ for any integer k because adding multiples of 2π to the angle 0 does not change the value of e^iθ due to the periodicity of sine and cosine functions.

$z^n = e^{i·2πk}$ ⇒[Solving for z:] $z = e^{\frac{i·2πk}{n}}$ for k = 0, 1, 2, ···, n -1. There are n distinct solutions corresponding to k = 0 through k = n-1. Using Euler’s formula, each root can be expressed as: $z_k = cos(\frac{i·2πk}{n})+isin(\frac{i·2πk}{n})$ where k = 0, 1, 2, ···, n -1.

Conclusion: The n-th roots of unity are the solution to the equation zⁿ = 1, where z is a complex number. These roots are given by the formula: $z_k = e^{\frac{2πik}{n}} = cos(\frac{2πk}{n}) + isin(\frac{2πk}{n})$ where k = 0, 1, 2, ···, n-1 and:

• $e^{\frac{2πik}{n}}$ represents the exponential form of complex numbers, derived from Euler’s formula e^iθ = cos(θ) + isin(θ).
• k is the index that determines the specific root.
• These roots are evenly spaced around the unit circle in the complex plane at angles that are multiples of $\frac{2π}{n}$

Examples of N-th Roots of Unity

• Square Roots of Unity (n = 2): z² = 1.

k = 0, $z_0 = e^{\frac{2πi·0}{2}}=e^0=1$. k = 1, $z_1 = e^{\frac{2πi·1}{2}}=e^{πi} = cos(π)+isin(π) = -1 + i·0 = -1.$

The square roots of unity are 1 and −1, lying on opposite ends of the real axis on the unit circle.

• Fourth root of Unity (n = 4): z⁴ = 1.

k = 0, $z_0 = e^{\frac{2πi·0}{4}} = e^0 = 1$. k = 1, $z_1 = e^{\frac{2πi·1}{4}} = e^{\frac{πi}{2}} = cos(\frac{π}{2})+isin(\frac{π}{2}) = 0 + i·1 = i$. k = 2, $z_2 = e^{\frac{2πi·2}{4}}=e^{πi} = cos(π)+isin(π) = -1 + i·0 = -1$. k = 3, $z_3 = e^{\frac{2πi·3}{4}} = e^{\frac{3πi}{2}} = cos(\frac{3π}{2})+isin(\frac{3π}{2}) = 0 + i·(-1) = -i$

The fourth roots of unity are 1, i, −1, and -i, which are positioned at equal angles of ^π⁄₂ radians around the unit circle.

In the complex plane, the n-th roots of unity are arranged as the vertices of a regular n-sided polygon inscribed in the unit circle. Each vertex corresponds to one of the roots, and the angle between any two consecutive roots is $\frac{2π}{n}$ radians (Refer to Figure iv for a visual representation and aid in understanding it).

Properties of N-th Roots of Unity:

1. Distinct Roots: All n roots z₀, z₁, ··, z_n-1 are distinct.
2. Equally Spaced on the Unit Circle: Each root is obtained by rotating the previous one by $\frac{2π}{n}$ radians, meaning they are evenly distributed around the circle.
3. Product of Roots. The product of any two n-th roots of unity is also an n-th root of unity: $z_k·z_m = e^{\frac{2πik}{n}}e^{\frac{2πim}{n}} = e^{\frac{2πi(k+m)}{n}} = z_{(k + m)~ mod~ n}$
4. Inverse of Each Root. Each n-th root of unity has an inverse, which is its complex conjugate: $z_k^{-1} = \bar z_k = e^{-\frac{2πik}{n}} = z_{n-k}$.
5. Sum of All N-th Roots of Unity. The sum of all n-th roots of unity is zero: $\sum_{k=0}^{n-1} z_k = 0$.

   This property holds because the roots are symmetrically distributed around the origin in the complex plane, so their vector sum cancels out.

Bibliography

1. NPTEL-NOC IITM, Introduction to Galois Theory.
2. Algebra, Second Edition, by Michael Artin.
3. LibreTexts, Calculus and Calculus 3e (Apex). Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
4. Field and Galois Theory, by Patrick Morandi. Springer.
5. Michael Penn, and MathMajor.
6. Contemporary Abstract Algebra, Joseph, A. Gallian.
7. YouTube’s Andrew Misseldine: Calculus. College Algebra and Abstract Algebra.
8. MIT OpenCourseWare [18.03 Differential Equations, Spring 2006], YouTube by MIT OpenCourseWare.
9. Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences.