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;
Conway 17.

@ __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;
Marshall 19.

@ __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 30 aug 2019