Analysis

The study of limits, continuity, differentiation, and integration.

Real Analysis

• Sequence — Function from N to R
• Convergence — lim a_n = L
• Divergence — Not convergent
• Limit Superior — limsup
• Limit Inferior — liminf
• Cauchy Sequence — Self-test for convergence
• Series — Sum of a sequence
• Partial Sum — s_n = a_1 + ... + a_n
• Geometric Series — Sum of ar^n
• Harmonic Series — Sum of 1/n
• p-Series — Sum of 1/n^p
• Comparison Test — For convergence
• Ratio Test — For convergence
• Root Test — For convergence
• Alternating Series Test — For convergence
• Absolute Convergence — Sum of |a_n| converges
• Conditional Convergence — Converges but not absolutely
• Function Limit — lim f(x) = L
• Continuity — lim f(x) = f(a)
• Discontinuity — Not continuous
• Uniform Continuity — δ doesn't depend on point
• Differentiability — f'(a) exists
• Derivative — Rate of change
• Mean Value Theorem — f'© = (f(b)-f(a))/(b-a)
• Rolle's Theorem — Special case of MVT
• Taylor Theorem — Polynomial approximation
• Taylor Series — Infinite Taylor polynomial
• Power Series — Sum a_n x^n
• Radius of Convergence — Where power series converges
• Riemann Integral — Upper and lower sums
• Fundamental Theorem of Calculus — Connects derivative and integral
• Integration by Parts — Technique
• Substitution Rule — u-substitution
• Improper Integral — Infinite bounds

Complex Analysis

• Complex Number (Analysis) — a + bi in analysis context
• Complex Differentiability — Holomorphic functions
• Cauchy-Riemann Equations — Test for holomorphic
• Contour Integral — Integral along a path
• Cauchy's Theorem — Integral around closed path
• Residue — Coefficient of 1/(z-a)
• Residue Theorem — Sum of residues