Real and Complex Analysis

In General > s.a. Calculus; functional analysis; operator theory; integration; series; vector calculus.
* Idea: Real/complex analysis is the mathematical theory of functions of a real/complex variable.
@ General books: Choquet 69; Pólya & Szegő 72; Gleason 66/91; Wong 10 [applied].
@ Real analysis, II: Pons 14 [II]; Laczkovich & Sós 15; Jacob & Evans 15.
@ Real analysis, advanced: Bourbaki 58; Royden 63; Knapp 05 [2 vol, basic + advanced]; Trench 03 (updated 12).
@ Complex analysis: Bochner & Martin 48; Ahlfors 53; Pólya & Latta 74; Priestley 03; Sasane & Sasane 13 [friendly]; Chakraborty et al 16.
@ Non-linear analysis: Rassias 86 [fixed point and bifurcation theory, non-linear operators].
@ Related topics: Rockafellar 68 [convex]; Sirovich 71, de Bruijn 81 [asymptotic]; Klebaner 12 [stochastic calculus]; > s.a. Cauchy Theorem; Cauchy-Riemann; Convex Functions.
> Related topics: see connection; Covariant, Fréchet, and Weak Derivative; differential equations; integral equations.

"Less Than Continuous" Functions > s.a. distributions; path integrals ["jaggedness" of paths]; Semicontinuity; Derivative [subdifferential].
* Types: The worst case is when a function does not have a limit along some or all directions at a point p.
* Direction-dependent limit: The limit of a function f along any curve γ passing through p exists and depends only on the tangent vector v to γ at p; We call this limit $$\cal F$$(v).
* Regular direction-dependent limit: The direction-dependent limit $$\cal F$$(v) of the function f admits derivatives to all orders with respect to v, and the operation of taking the limit of f along γ commutes with taking these derivatives.
* Itô calculus: A generalized form of calculus that can be applied to non-differentiable functions, and is one of the branches of stochastic calculus; Applications: It can be used to derive the general form of the Fokker-Planck equation; > s.a. Wikipedia page.

Continuity Classes of Functions > s.a. Hölder and Lipschitz condition.
* Types: A map f : XY between two differentiable manifolds can be
- C0: f is continuous.
- C>0: f is C0 and its derivatives have regular direction-dependent limits.
- C1/2: Δf/(Δx)1/2 approaches a finite limit as Δx → 0.
- C1–: f satisfies the Lipschitz condition.
* Conditions involving derivatives:
- Cr, for some integer r: f is continuously differentiable up to the r-th order derivatives.
- Cr0: f is Cr and has compact support.
- Cr: f is Cr–1 and its (r–1)-th derivatives are locally Lipschitz functions.
- C>r: f is Cr and its (r+1)-th derivatives have regular direction-dependent limits.
- C: f is infinitely differentiable.
- Cω: f is analytic.
* Remark: An example of a function which is C but not Cω at x = 0 is f(x) = e–1/x; C submanifolds of a manifold can merge, Cω ones can't.

Special Types and Generalizations > s.a. functions; Expansion of a Function; Special Functions; Takagi Function; Weierstraß Functions.
@ Examples: Gelbaum & Olmsted 64 [counterexamples]; Ramsamujh CJM(89) [nowhere differentiable C0]; Oldham et al 08 [atlas of functions].
@ Generalizations: Shale JFA(74) [over discrete spaces]; Heinonen BAMS(07) [non-smooth calculus]; Smirnov a1009-proc [possible discretizations]; > s.a distribution; fractional calculus; non-standard analysis.