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].

@ __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.

> __Online resources__:
see Wikipedia page.

> __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* : *X* → *Y* between
two differentiable manifolds can be

- C^{0}: *f* is continuous.

- C^{>0}: *f* is
C^{0} and its derivatives have regular direction-dependent limits.

- C^{1/2}: Δ*f*/(Δ*x*)^{1/2} approaches
a finite limit as Δ*x* → 0.

- C^{1–}: *f* satisfies
the Lipschitz condition.

* __Conditions involving derivatives__:

- C^{r},
for some integer *r*: *f* is
continuously differentiable up to the *r*-th order derivatives.

- C^{r}_{0}: *f* is C^{r} and has compact support.

- C^{r–}: *f* is
C^{r–1} and its (*r*–1)-th
derivatives are locally Lipschitz functions.

- C^{>r}: *f* is
C^{r} and its (*r*+1)-th derivatives have regular direction-dependent limits.

- C^{∞}: *f* is infinitely differentiable.

- C^{ω}: *f* is analytic.

* __Remark__: An example
of a function which is C^{∞} but
not C^{ω} at
*x* = 0 is *f*(*x*) = e^{–1/x};
C^{∞} submanifolds
of a manifold can merge, C^{ω} 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 C^{0}]; 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.

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send feedback and suggestions to bombelli at olemiss.edu – modified
18 jan 2016