Formulas: Higher mathematics

Limits and series

• limx→0 (sin x) / x = 1

• limx→∞ (1 + 1 / x)x = e

• Geometric series sum: Sn = a1(qn − 1) / (q − 1)


Logarithms

• logab = ln b / ln a

• loga(xy) = logax + logay

• loga(x / y) = logax − logay


Integrals and derivatives

• (xn)′ = n xn−1

• (ex)′ = ex

• ∫ xn dx = xn+1 / (n + 1) + C, n ≠ −1

• ∫ dx / x = ln|x| + C

• y = f(x0) + f′(x0)(x − x0) — tangent line


Complex numbers and Taylor

• z = a + bi, |z| = √(a² + b²)

• z̄ = a − bi, arg z = arctan(b / a)

• eiφ = cos φ + i sin φ

• f(x) ≈ f(a) + f′(a)(x − a) + f′′(a)(x − a)² / 2!


Differential equations

• y′ = ky ⇒ y = Cekx

• y′ + P(x)y = Q(x) — first-order linear form

• y′ = f(x)g(y) ⇒ dy / g(y) = f(x) dx