Complex analysis lemmas

In mathematics, the Calderón–Zygmund lemma is a fundamental result in Fourier analysis, harmonic analysis, and singular integrals.
It is named for the mathematicians Alberto Calderón and Antoni Zygmund.

In mathematics, Itô's lemma or Itô's formula is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process.
It serves as the stochastic calculus counterpart of the chain rule.
It can be heuristically derived by forming the Taylor series expansion of the function up to its second derivatives and retaining terms up to first order in the time increment and second order in the Wiener process increment.
The lemma is widely employed in mathematical finance, and its best known application is in the derivation of the Black–Scholes equation for option values.

In mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact.
Precisely, it states that every closed p-form on an open ball in Rn is exact for p with nowrap>1 ≤ p ≤ n.
The lemma was introduced by Henri Poincaré in 1886.

In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself.
The lemma is less celebrated than deeper theorems, such as the Riemann mapping theorem, which it helps to prove.
It is, however, one of the simplest results capturing the rigidity of holomorphic functions.