Complex Numbers and Euler's Formula | MIT Differential Equations

Trigonometry begins with right triangle ratios: sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), tangent (opposite/adjacent). The unit circle extends these to all angles, where cos(theta) = x-coordinate and sin(theta) = y-coordinate. Key angles (30°, 45°, 60°) have special values. Degrees convert to radians via π/180. Function graphs: sine/cosine oscillate between -1 and 1 with period 2π; tangent has period π with vertical asymptotes. Polar coordinates use (r, theta) instead of (x, y), with conversion formulas x = r cos(theta), y = r sin(theta).

The unit circle is centered at (0,0) with radius 1, where x = cos(θ) and y = sin(θ). Conversions: degrees to radians multiply by π/180, radians to degrees multiply by 180/π. Key special triangles: 30-60-90 (side opposite 30° is half hypotenuse, opposite 60° is half hypotenuse × √3), 45-45-90 (legs equal, hypotenuse = leg × √2). Key coordinates: (1,0) at 0°, (0,1) at 90°, (-1,0) at 180°, (0,-1) at 270°. Quadrant signs: QI all positive, QII sin/csc positive, QIII tan/cot positive, QIV cos/sec positive. Trigonometric functions: sin(θ) = y, cos(θ) = x, tan(θ) = y/x, cot(θ) = x/y, sec(θ) = 1/x, csc(θ) = 1/y. For π/6: (√3/2, 1/2), sin=1/2, cos=√3/2, tan=√3/3, csc=2, sec=2√3/3, cot=√3. For 5π/4: (-√2/2, -√2/2), sin=-√2/2, cos=-√2/2, tan=1, csc=-√2, sec=-√2, cot=1.

Polar coordinates use (r, θ) instead of (x, y) to locate points. The pole is the origin, the polar axis extends from it, and concentric circles represent radius values. The radius r measures distance: positive r moves outward along the angle, negative r moves in the opposite direction (180° away). The angle θ measures direction: positive angles go counterclockwise, negative angles go clockwise. One full rotation equals 360° or 2π radians. A single point has infinitely many representations: (r, θ + 2πn), (-r, θ + π + 2πn), and (r, -θ + 2πn). Converting to rectangular: x = r cos(θ), y = r sin(θ). Converting from rectangular: r = √(x² + y²), θ = arctan(y/x). Key tangent values: tan(π/6) = √3/3, tan(π/4) = 1, tan(π/3) = √3.

Trigonometry studies right-angled triangles where the three main functions are defined: Sine (sin) = opposite/hypotenuse, Cosine (cos) = adjacent/hypotenuse, and Tangent (tan) = opposite/adjacent. The Pythagorean identity sin²(α) + cos²(α) = 1 is fundamental. Reciprocal functions include cosecant (csc), secant (sec), and cotangent (cot). Trigonometry extends to the unit circle (radius 1 centered at origin) for defining functions for any angle. The unit circle uses Cartesian coordinates where x = cos(α) and y = sin(α). Two angle measurement systems exist: degrees (360° full circle) and radians (2π radians full circle). Conversion: 180° = π radians. Radians are preferred in physics and higher mathematics for simplifying circular motion calculations.

This lesson covers converting Cartesian coordinates (x, y) to polar coordinates (r, θ). The radius r is calculated using the Pythagorean theorem: r = √(x² + y²). The angle θ is found using cos(θ) = x/r and sin(θ) = y/r. The instructor demonstrates with the point (1/2, -√3/2), showing how to calculate r = 1 and θ = -π/3. Understanding the unit circle is essential for correctly identifying angles from trigonometric values.

Exponential functions are among the most important units in mathematics, alongside logarithmic functions. Euler discovered that the function satisfying f'(x) = f(x) and f(0) = 1 is the exponential function, denoted as e^x. Euler's number e (approximately 2.71828) is defined as e^1. The exponential function follows key algebraic properties: product rule (e^x × e^y = e^(x+y)), reciprocal rule (e^x × e^(-x) = 1), quotient rule (e^x / e^y = e^(x-y)), and power rule ((e^x)^n = e^(n×x)). These properties mirror those of powers with the same base in algebra and are essential for simplifying exponential expressions.

Euler's constant (e ≈ 2.718) is a mathematical constant that appears in many areas of mathematics and physics. Properties: e^0 = 1, e^∞ = ∞, e^(-∞) = 0. Important mathematical constants to memorize: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, π ≈ 3.1416, π² ≈ 9.87, g (acceleration due to gravity) ≈ 9.8 m/s². These are frequently used in physics calculations.

Leonhard Euler was the first mathematician to publish e in his book 'Mechanica', naming it after himself using the letter 'e'. Euler's publication established e as a standard mathematical constant. The number e has two key properties: it is an irrational number (cannot be expressed as a fraction) and a transcendental number (cannot be a root of any polynomial equation with rational coefficients). Unlike √2, which is irrational but not transcendental (as it satisfies x² - 2 = 0), e cannot be expressed as the solution to any polynomial equation with rational coefficients.

Euler's number (e ≈ 2.71828) is a fundamental constant in mathematics, arising from compound interest calculations. As compounding frequency approaches infinity, the growth factor approaches e. Like Pi, e is transcendental and irrational. Euler's identity e^(iπ) + 1 = 0 connects five fundamental constants (e, i, π, 1, 0) in one elegant equation, often considered the most beautiful in mathematics.

Leonhard Euler formalized e's properties fifty years after Bernoulli's discovery. He named it 'e' using the next available letter, creating a beautiful coincidence honoring his work. Euler proved e is irrational through continued fractions: e = 2 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + 1/(4 + ...))))), showing it never terminates. He also derived the series expansion e = 1 + 1/1! + 1/2! + 1/3! + ..., calculating it to 18 decimals. Most significantly, Euler revealed that y = e^x uniquely satisfies having equal value, derivative, and area under the curve at every point—making it the natural language of calculus for modeling growth and change.

This section covers the derivation of Maclaurin series for three fundamental functions. For e^x, the series is sum of x^n/n! from n=0 to infinity, derived because all derivatives equal e^x and evaluate to 1 at x=0. For sin(x), only odd powers appear because even-order derivatives at x=0 are zero, yielding sum of (-1)^n * x^(2n+1)/(2n+1)!. For cos(x), only even powers appear because odd-order derivatives at x=0 are zero, yielding sum of (-1)^n * x^(2n)/(2n)!. These series form the foundation for proving Euler's formula.

The proof of Euler's formula relies on the Maclaurin series expansions of e^x, cos(x), and sin(x). The Maclaurin series for e^x is 1 + x + x²/2! + x³/3! + x⁴/4! + ..., for cos(x) is 1 - x²/2! + x⁴/4! - x⁶/6! + ..., and for sin(x) is x - x³/3! + x⁵/5! - x⁷/7! + .... These series provide the foundation for deriving Euler's formula by substituting ix for x in the exponential series.

The Maclaurin series expansions for these functions are: cos(θ) = 1 - θ²/2! + θ⁴/4! - θ⁶/6! + ...; sin(θ) = θ - θ³/3! + θ⁵/5! - θ⁷/7! + ...; e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + ... These infinite series representations allow us to prove Euler's formula by substituting x = iθ into the exponential series.

The Maclaurin series for e^x is: e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + ... with all coefficients equal to 1. For sin(x): sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + ... containing only odd powers with alternating signs. For cos(x): cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ... containing only even powers with alternating signs. Substituting x = iθ into e^x's series yields: e^(iθ) = 1 + iθ - θ²/2! - iθ³/3! + θ⁴/4! + iθ⁵/5! - ... Grouping terms: e^(iθ) = (1 - θ²/2! + θ⁴/4! - ...) + i(θ - θ³/3! + θ⁵/5! - ...) = cos(θ) + i·sin(θ). This proves Euler's Formula, revealing the deep connection between exponential and trigonometric functions in complex analysis.

The Taylor series expansions around x = 0 (Maclaurin series) are: e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + ..., sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + ..., cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ... Substituting x = iθ into the exponential series gives e^(iθ) = 1 + iθ - θ²/2! - iθ³/3! + θ⁴/4! + iθ⁵/5! - ... = (1 - θ²/2! + θ⁴/4! - ...) + i(θ - θ³/3! + θ⁵/5! - ...) = cosθ + i sinθ. This proves Euler's formula and demonstrates the deep connection between exponential and trigonometric functions.