#C07 · Calculus & Analysis · Lv.4View in 3D Mind-Map

Euler's Formula & Euler's Identity

Leonhard Euler · 1748 CE

e^{i\theta} = \cos\theta + i\sin\theta,\quad e^{i\pi} + 1 = 0

\cos\theta = \frac{e^{i\theta}+e^{-i\theta}}{2},\quad \sin\theta = \frac{e^{i\theta}-e^{-i\theta}}{2i}

Unites exponential growth e^x and circular trigonometry cos θ + i sin θ in the complex plane—celebrated as the jewel of mathematics and the core of Fourier analysis and quantum mechanics.

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Phase angle θ°

e^{i(180^\circ)} = \cos(180^\circ) + i\sin(180^\circ) = -1.000 + (0.000)i

실수부 Re = cos θ

-1.0000

허수부 Im = sin θ

0.0000

|e^(iθ)| 단위원 크기

1.0000

Prerequisite Path (16 STEP)

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Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 2#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 3#G01Pythagorean Theorem

a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}

STEP 4#G03Pi, Radian Measure & Circular Sector

l = r\theta

STEP 5#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 6#A04Quadratic Formula & Discriminant

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

STEP 7#A06Logarithm Laws & Change of Base

\log_a(xy) = \log_a x + \log_a y

STEP 8#A08Binomial Theorem & Pascal's Triangle

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

STEP 9#A07Arithmetic & Geometric Series Sum

S_n = \frac{a(1 - r^n)}{1 - r}

STEP 10#C02Derivative Definition & Instantaneous Rate

f'(x) = \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

STEP 11#G04Unit Circle Trigonometry & Identities

\cos^2\theta + \sin^2\theta = 1

STEP 12#C01Limits, Continuity & Euler's Number e

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828

STEP 13#C04Fundamental Theorem of Calculus (FTC)

\int_a^b f(x)\

STEP 14#A09Complex Numbers & Polar Plane

z = a + bi = r(\cos\theta + i\sin\theta)

STEP 15#C06Taylor & Maclaurin Series Expansion

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

STEP 16#C07Euler's Formula & Euler's Identity

e^{i\theta} = \cos\theta + i\sin\theta

🚀 Next Unlocks (Derived Formulas) (3)

#C09Fourier Series & Fourier Transform (FFT)#C10Differential Equations & Harmonic Oscillator #L10Quaternions & 3D Spherical Interpolation (SLERP)

Geometric & Intuitive Insight

e^x 의 테일러 급수에 x 대신 iθ 를 넣으면, 짝수 차수 항들은 i²=-1 에 의해 부호가 교차하는 cos θ 급수가 되고, 홀수 차수 항들은 i로 묶여 sin θ 급수가 되어 두 세계가 완벽히 하나로 합쳐집니다!

Step-by-Step Derivation

Step 1.1단계: 매클로린 급수 e^(iθ) = 1 + iθ + (iθ)²/2! + (iθ)³/3! + (iθ)⁴/4! + … 를 전개합니다.
Step 2.2단계: i² = -1, i³ = -i, i⁴ = 1 을 대입해 실수부와 허수부로 묶습니다 → (1 - θ²/2! + θ⁴/4! - …) + i(θ - θ³/3! + θ⁵/5! - …)
Step 3.3단계: 괄호 안이 각각 cos θ 와 sin θ 의 매클로린 급수이므로 e^(iθ) = cos θ + i sin θ (θ=π 대입 시 e^(iπ) + 1 = 0)

AI & Coding Real-World Use

LLM Rotary Position Embedding (RoPE e^(imθ)), FFT complex basis, and quantum phase gates

#LLM RoPE e^(imθ)#Fourier Basis#Quantum Phase

const eulerCis = (theta: number) => ({ re: Math.cos(theta), im: Math.sin(theta) });

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