Real
and Complex Analysis| Real
and Complex Analysis |
In General > s.a. Calculus; functional
analysis.
> 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.
* 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 
passing
through p exists and depends only on the tangent
vector v to
at p;
We call this limit 
(v).
* Regular direction-dependent
limit: The direction-dependent limit
(v) of the function f admits
derivatives to all orders wrt v,
and the
operation of taking the limit of f along commutes
with taking these derivatives.
* Ito calculus: A generalized form of calculus that can be applied
to non-differentiable functions; Applications: Can be used to derive
the general form of the Fokker-Planck equation.
Continuous Functions > s.a. Hölder and
Lipschitz condition.
* Types: A map f : X → Y 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: 
f/(x)1/2 approaches
a finite limit as
x → 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.
- Cinfty: f is
infinitely differentiable.
- Comega: f is
analytic.
* Remark: An example
of a function which is Cinfty but
not Comega at
x = 0 is f(x) = e–1/x;
Cinfty submanifolds
of a manifold can merge, Comega ones
can't.
Special Types and Generalizations > s.a. functions; Expansion
of a Function; Weierstraß Functions.
@ Examples: Gelbaum & Olmsted 64 [counterexamples]; Ramsamujh CJM(89)
[nowhere differentiable C0]; > s.a. Special
Functions.
@ Generalizations: Shale JFA(74)
[over discrete spaces]; Heinonen BAMS(07)
[non-smooth calculus]; > s.a distribution; non-standard
analysis.
References > s.a. Cauchy
Theorem; Cauchy-Riemann;
Convex Functions; integration; series;
vector calculus.
@ General: Choquet 69; Polya & Szegö 72; Gleason 66/91.
@ Real analysis: Bourbaki; Royden 63; Knapp 05 [2 vol, basic + advanced].
@ Complex analysis: Bochner & Martin 48; Ahlfors 53; Polya & Latta
74.
@ Non-linear analysis: Rassias 86 [fixed point and bifurcation theory,
non-linear operators].
@ Related topics: Rockafellar 68 [convex]; Sirovich 71, de Bruijn 81 [asymptotic].
Fractional Derivatives > s.a. differential
equations; differential
forms; gauge transformations; vector
calculus.
* Idea: Developed by
Riemann-Liouville; They are related to (multi)fractals, and have applications
in time series, kinetics of chaotic systems, polymer science, biophysics; Can
behave as dissipative terms in dynamical systems.
@ General references: in Gel'fand & Shilov 64; in Bateman 54, v2,
chXIII;
Nigmatullin TMP(92);
Hilfer ed-00 [intro and applications].
@ Interpretation: Rutman TMP(95); Podlubny FCAA(02)m.CA/01 [and
integration]; Fa PhyA(05)
[and dissipation, falling body].
@ Fractional Schrödinger equation: Laskin qp/02;
Naber
JMP(04)mp [t-fractional];
Herrmann mp/05;
Guo & Xu JMP(06)
[examples].
@ Related topics: Kobelev m.CA/00 [on
multifractal sets], ht/00 [and
multifractal
spacetime]; Muslih & Rabei MPLA(05)
[mechanical systems, quantization]; Baleanu & Muslih PS(05)
[fractional Lagrangian
for
field theory]; Muslih & Baleanu NCB(05)
[in quantum field theory]; Muslih et al PS(06)
[Hamiltonian formulation and path integral quantization]; Atanackovic et al JPA(08)
[Euler-Lagrange
equations]; > s.a. hamilton-jacobi.
> Other applications:
see brownian
motion; classical systems [non-conservative]; description
of chaos; diffusion; hamiltonian
systems; lagrangian systems; variational
principles.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
12 jun 2008