Complex Numbers and Differential Equations

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Recall. Algebraic and differential equations.

An algebraic equation is a mathematical statement asserting that two algebraic expressions are equal.. These expressions are constructed using:

1. Dependent and independent variables. Variables are symbols representing unknown quantities. Variables can be:
   Independent Variables: Variables or input that can be chosen (or change) freely.
   Dependent Variables: Variables that depend on the independent variables.
2. Constants. Fixed numerical values that do not change, e.g., 2, π, 7.
3. Algebraic operations. Operations such as addition, subtraction, multiplication, division, exponentiation, powers, and root.

An algebraic equation typically has the following general form: Expression₁ = Expression₂, where the two expressions are set equal to each other.

Examples: y = 2x + 4 (simple linear equation, represents a straight line with a slope of 2 and a y-intercept of 4), ax² + bx + c = 0 (quadratic equation, it represents a parabola), $ln(xy + t) = z·sin(\sqrt{x})$.

Definition. A differential equation is an equation that involves one or more functions and their derivatives. It relates the function itself (the dependent variable), its derivatives with respect to one or more independent variables (rates of change), and the independent variable(s) themselves, e.g., $\frac{dy}{dx} = 3x +5y, y' + y = 4xcos(2x), \frac{dy}{dx} = x^2y+y, etc.$

If an algebraic equation is a “snapshot” of a relationship, a differential equation is the “instruction manual” for how that relationship moves and changes.

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

• Dependent variables: The function y (or y(x)) we are trying to find or solve for. Variable(s) (y) that depend on one or more other variables.
• Independent variables: Variable(s) (x) upon which the dependent variables depend and we differentiate with respect to (often time or space).
• Derivatives: Expressions like $\frac{dy}{dx}$ or y’ that represent the rate at which the dependent variable (y) change with respect to the independent variable (x).

First-Order Linear Ordinary Differential Equations (ODEs)

First-order linear differential equations are fundamental tools in mathematical modeling across various disciplines, including physics, engineering, biology, and economics. They are used to describe dynamic systems where the rate of change of a quantity depends linearly on the quantity itself and, frequently, on external inputs.

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 first derivative of y with respect to x.
• a(x), b(x), and c(x) are known continuous functions of x.

These equations are termed “linear” because the unknown function y and its derivative y’ strictly appear to the first power. They are never multiplied together, squared, or composed inside nonlinear functions (like sin(y) or $e^y$).

If the forcing 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. If $c(x) \ne 0$, it is non-homogeneous.

The Standard Linear Form

To standardize the equation and prepare it for universal solution methods, we divide both sides of the general form by a(x), assuming a(x) ≠ 0: y’ + p(x)y = q(x) where $p(x)=\frac{b(x)}{a(x)}\text{ and }q(x) = \frac{c(x)}{a(x)}$ This is the standard linear form of a first-order differential equation, which is more convenient for applying solution methods, such as the integrating factor method.

Complex Numbers

Complex numbers are a fundamental extension of the real numbers and play a crucial role in mathematics, physics, engineering, signal processing, quantum mechanics, and computer graphics.

Definition. 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: $z·\bar z = (a + bi)(a -bi) = a^2 + b^2 = |z|^2$, 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 Form and Euler’s Formula

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 $\theta = arg(z) = tan^{-1}(\frac{b}{a})$. It is the angle between the positive real axis and the line representing the complex number. The angle $\theta$ should be determined in the correct quadrant, considering the signs of a and b.

The polar form of a complex number is: $z = a + bi = r\cos(\theta) + r\sin(\theta)i = r(\cos(\theta) + i\sin(\theta))$ =[This can be written compactly using Euler’s formula] $re^{i\theta}$.

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 $e^{i\theta} = \cos(\theta) + i\sin(\theta)$. 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. For a nonzero complex number 𝑎, define $\boxed{a^z = e^{z\log(a)}}$ where $\log(a)$ is the principal logarithm unless stated otherwise. Under this convention, the familiar rules hold:

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: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$

Complex exponentiation is multi‑valued unless you choose a branch of the logarithm. Because $log(a) = ln|a| + i(arg(a) + 2\pi k)$, the expression $a^z = e^{z\log(a)}$ has infinitely many values unless you fix a branch, e.g., $(-1)^{1/2}$ has two valued i and -i.

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: $z_1·z_2 = r_1r_2e^{iθ_1}·e^{iθ_2}=[\text{Laws of Exponents. Product Rule.}]r_1r_2e^{iθ_1+iθ_2}=\boxed{r_1r_2e^{i(θ_1+θ_2)}}$. 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: $\frac{z_1}{z_2} = \frac{r_1e^{iθ_1}}{r_2e^{iθ_2}} =[\text{Laws of Exponents. Product Rule.}] \frac{r_1}{r_2}e^{iθ_1-iθ_2}=\boxed{\frac{r_1}{r_2}e^{i(θ_1-iθ_2)}}$. In words, moduli divide and arguments subtract.

Differentiation and Integration

Differentiation of Complex Functions A complex‑valued function of a real variable 𝑡 can always be written as: 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. Because differentiation is linear, the derivative of y(t) with respect to t is obtained by differentiating the real and imaginary parts separately: $\frac{d}{dt}y(t) = \frac{d}{dt}[u(t)+iv(t)] = \frac{du}{dt}+i\frac{dv}{dt}$. Symbolically, D(u + iv)= Du + iDv. This rule is valid because differentiation with respect to a real variable is a real‑linear operator.

Example: Differentiating the Complex Exponential Function.

Consider the function y(t) = e^it.

$$ \begin{aligned} \frac{d}{dt}e^{it} &=[\text{Using Euler's formula}] \frac{d}{dt}(cos(t) + isin(t)) \\[2pt] &=[\text{Differentiation Rule}] \frac{d}{dt}(cos(t)) + i\frac{d}{dt}(sin(t)) \\[2pt] &=-sin(t) + icos(t) = i(cos(t) + isin(t)) \\[2pt] &=[\text{Using Euler's formula}] ie^{it}. \end{aligned} $$

This 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 corresponds to a 90° counterclockwise rotation in the complex plane. So the derivate of $e^{it}$ is the same vector roated by 90. This property is fundamental in solving differential equations involving complex exponentials.

Integration follows similar principles, integrating the real and imaginary parts separately. $\int y(t)dt = \int u(t)dt + \int v(t)dt$.

Consider $\int e^{it}dt = \int (cos(t)+isin(t))dt = sin(t)-icos(t)+C =[\text{Factor out -i}] -i(cos(t)+sin(t)i)+C =-ie^{it}+C$.

A complex‑valued function of a real variable behaves exactly like a vector‑valued function in $\mathbb{R}^2$. Differentiation and integration are done componentwise because:

• u(t) and v(t) are real functions,
• the imaginary unit 𝑖 is just a constant scalar,
• no complex‑analytic conditions (like Cauchy–Riemann) are required.

This is not the same as differentiating a function $f: \mathbb{C} \to \mathbb{C}$ with respect to a complex variable 𝑧. There, differentiability is much more restrictive and leads to analyticity.

Solving Differential Equations Involving Complex Numbers

Let’s explore the solution and implication of a fundamental linear differential equation where the coefficient is the imaginary unit.

Consider the first-order homogeneous linear ordinary differential equation (ODE): $\frac{dy}{dt} = iy$ where:

• y(t) is a complex-valued function of a real variable t. y is a function of t, and
• i is the imaginary unit.

We seek the specific solution given the initial condition: y(0) = 1.

1. Finding the General Solution.

Because this equation is separable, we can solve it by isolating variables. Assuming $y \ne 0$, divide by y and multiply by dt to separate the 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.

(Note: While technically ln∣y∣ for real variables, in the complex domain, we proceed with the complex logarithm, where the constant C absorbs necessary adjustments).

To solve for y, exponentiate both sides: $y = e^{it+C} =[\text{Using the laws of exponents}] e^Ce^{it}$.

Since C is an arbitrary constant of integration, $e^C$ is also an arbitrary constant. Let us define a new constant, K = e^C. Therefore, the general solution is $y(t) = Ke^{it}$.

2. Applying the initial condition.

We are given that when t = 0, y = 1. We substitute these values into the general solution to find the specific value of K: y(0) = Ke^i·0 = K·1 = K = 1. Substituting K = 1 back into the general solution gives us the unique final solution: $\boxed{y(t) = e^{it}}$

3. Verification.

To confirm this solution is correct, we must verify that it satisfies both the original differential equation and the initial condition.

• Check initial condition: $y(0)=e^{i·0} = cos(0) + isin(0) = 1 + i·0 = 1$. This matches the given condition.
• Check differential equation: Using the chain rule to differentiate $y(t) = e^{it}$ with respect to t: $\frac{d}{dt}y(t) = \frac{d}{dt}e^{it} = ie^{it} = i(y(t))$.
• This confirms that y(t) = e^it satisfies both the differential equation and the initial condition.

Exponential Expressions with Complex Exponents

The solution to the differential equation above relies entirely on the concept of complex exponents. Understanding how to interpret terms like e^it is crucial for working with complex functions and differential equations in physics and engineering.

General Expression and Deconstruction

Consider the general exponential expression e^a+ib where the exponent is a complex number z = a +ib, and a and b are real numbers, a, b ∈ ℝ.

Using the standard properties of exponents, this expression can be rewritten as a product of a real exponential and a complex exponential: e^a+ib = e^a·e^ib where e^a is a standard, positive real number representing the magnitude (or modulus) of the complex number and e^ib is a purely complex exponential, representing the rotation in the complex plane.

Euler’s formula provides the bridge between complex exponentials and trigonometric functions: e^ib = cos(b) + isin(b).

Combining the expressions, the general complex exponential $e^{a+ib}$ can be written in its rectangular (Cartesian) form: $\boxed{e^{a+ib} = e^a(\cos(b) + i\sin(b))}$.

Geometric Interpretation

In the context of polar coordinates $(r, \theta)$:

• The magnitude $r = e^a$.
• The angle (argument) $\theta = b$ (measured in radians).

When viewed in the complex plane, the expression $e^{a+ib}$ representsa point in the complex plane at a distance of $e^a$ from the origin, rotated by an angle of b radians from the positive real axis.

If a = 0 (as in our differential equation solution $e^{it}$), the magnitude $e^0 = 1$. This implies that $e^{it}$ always lies on the unit circle in the complex plane, tracing a counter-clockwise path as t increases.

Bibliography

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