Determine a mathematically rigorous justification for a derivative and integration process leading to Euler's formula | Step-by-Step Solution

What We're Solving:

You're asking for a rigorous mathematical proof of Euler's famous formula: z = e^(iθ) = cos(θ) + i·sin(θ). This is one of the most beautiful relationships in all of mathematics, connecting exponential functions, trigonometry, and complex numbers!

The Approach:

There are several ways to prove Euler's formula, but I'll show you the most elegant approach using calculus. The key insight is to define a complex exponential function and show that it satisfies the same differential equation as trigonometric functions. If two functions satisfy the same differential equation with the same initial conditions, they must be identical!

Step-by-Step Solution:

Step 1: Define our complex exponential function Let f(θ) = e^(iθ) and explore what this means.

Step 2: Take the derivative Using the chain rule: f'(θ) = d/dθ[e^(iθ)] = i·e^(iθ) = i·f(θ)

This tells us something powerful: our function's derivative is i times itself!

Step 3: Set up the key relationship Now, let's consider what we know about sine and cosine derivatives:

• d/dθ[cos(θ)] = -sin(θ)
• d/dθ[sin(θ)] = cos(θ)

Step 4: Test our hypothesis Let's hypothesize that f(θ) = cos(θ) + i·sin(θ) and verify it satisfies our differential equation:

f'(θ) = d/dθ[cos(θ) + i·sin(θ)] f'(θ) = -sin(θ) + i·cos(θ) f'(θ) = i²·sin(θ) + i·cos(θ) [since i² = -1] f'(θ) = i(cos(θ) + i·sin(θ)) f'(θ) = i·f(θ) ✓

Perfect! This matches our result from Step 2.

Step 5: Check initial conditions At θ = 0:

• f(0) = e^(i·0) = e^0 = 1
• cos(0) + i·sin(0) = 1 + i·0 = 1 ✓

Both expressions equal 1 when θ = 0.

Step 6: Apply the uniqueness theorem Since both e^(iθ) and cos(θ) + i·sin(θ):

• 1. Satisfy the same differential equation: f'(θ) = i·f(θ)
• 2. Have the same initial condition: f(0) = 1

By the uniqueness theorem for differential equations, they must be the same function!

The Answer:

Therefore, we have rigorously proven that: e^(iθ) = cos(θ) + i·sin(θ)

This is Euler's formula, showing that the complex exponential function is intimately connected to circular motion and trigonometric functions.

Memory Tip:

Think of Euler's formula as describing a point moving around the unit circle in the complex plane. As θ increases, e^(iθ) traces out a circle with radius 1, where cos(θ) gives the x-coordinate and sin(θ) gives the y-coordinate. The exponential function e^(iθ) is just a compact way to represent this circular motion!

Bonus insight: When θ = π, you get the famous identity e^(iπ) + 1 = 0, connecting five of the most important numbers in mathematics: e, i, π, 1, and 0.

This proof beautifully demonstrates how calculus, complex analysis, and trigonometry work together. Keep exploring these connections – they're the heart of advanced mathematics!