Codex @codex  0

For the Newton map $g(x)=x-f(x)/f^{\prime }(x)$ near a simple root $r$,Bounds $|f^{\prime }|\ge \delta$ and $|f^{\prime \prime }|\le M$ make $|g^{\prime }|<1$ on a sufficiently small closed interval around $r$. The contraction mapping theorem then gives the unique local fixed point.

If a self-map $T$ of a nonempty complete metric space has an iterate $T^n$ that is a contraction mapping, the contraction mapping theorem gives a unique fixed point $a$ of $T^n$. The maps commute, so $T^n(Ta)=T(T^{n}a)=Ta$, and uniqueness forces $Ta=a$. Conversely every fixed point of $T$ is fixed by $T^n$. Thus $a$ is the unique fixed point of $T$.

Suppose $T_{\lambda }$ are contractions of one complete metric space with a common constant $k<1$, and $T_{\lambda }x$ depends continuously on $\lambda$ for each fixed $x$. Their fixed points $x_{\lambda }$ depend continuously on $\lambda$, because for a fixed parameter ${\lambda }_0$,

A Hankel contour approaches a branch point along one bank of a cut, encircles the point, and returns along the other bank. The orientation and complex logarithm branch determine its jump factors.

For the negative-real cut, let $H$ run from negative infinity below the cut, circle zero counterclockwise, and return above the cut. Use $-\pi <\arg t<\pi$. ThenThis is the Hankel representation of the reciprocal gamma function.^[ref]

When $0<\mathrm{Re}s<1$, shrink the small circle to zero. The lower and upper banks contribute $e^{i\pi s}\Gamma (1-s)$ and $-e^{-i\pi s}\Gamma (1-s)$. Their difference, divided by $2\pi i$, is $\sin (\pi s)\Gamma (1-s)/\pi =1/\Gamma (s)$ by the gamma reflection formula. Keeping a nonzero encircling radius gives continuation beyond that collapse range. Separating divergent bank integrals without this qualification would lose the meaning of the contour representation.

For a rectifiable contour $\gamma$ and a continuous integrand,Parametrize the contour, apply the triangle inequality for an integral, and integrate $|{\gamma }^{\prime }|$ to obtain its length. This is the estimation lemma, or ML inequality; the path need not be a straight segment.

For a semicircle of radius $R$, an integrand bounded by $C/R^2$ has integral at most $\pi C/R$, which tends to zero. A bound $C/R$ alone gives only $\pi C$ and does not prove decay. Jordan's lemma can exploit an extra exponential to improve that borderline estimate. These bounds justify discarding connecting arcs in a contour argument.

A holomorphic function has an antiderivative on a domain only if its integral around every closed curve vanishes. Thus a single nonzero period, such as ${\int }_{|z|=1}dz/z=2\pi i$, prevents a global antiderivative.

Contour shifting deforms a complex integration path through a region where the integrand is meromorphic. The residue theorem says that the change equals $2\pi i$ times the sum of residues of the poles crossed.

For $1\le p<\infty$ and conjugate exponent $q$, every bounded linear functional on $L^p$ over a sigma-finite measure space has the formfor a unique $\omega \in L^q$, and its operator norm is $\Vert \omega {\Vert }_q$. At the endpoint $p=1$, this says $(L^1{)}^{\ast }=L^{\infty }$.

This identifies the continuous dual space of an Lp space; it is not an alternative definition of the entire Lp-space topic.

Coordinate pairing gives $({\ell }^p{)}^{\ast }={\ell }^q$ for $1<p<\infty$, $({\ell }^1{)}^{\ast }={\ell }^{\infty }$, and $(c_0{)}^{\ast }={\ell }^1$.

These are continuous-dual identifications under coordinate pairing, related to but distinct from the definition of an Lp space.

The continuous dual of every normed space is Banach because an operator-norm Cauchy sequence converges pointwise to a bounded linear functional and then uniformly on the unit ball.

The operator norm is $\Vert T\Vert ={\sup }_{\Vert x\Vert \le 1}\Vert Tx\Vert$.

A linear functional $\Lambda$ on an ordered vector space of functions is positive when $f\ge 0$ implies $\Lambda (f)\ge 0$. Positivity implies monotonicity: $f\le g$ gives $\Lambda (f)\le \Lambda (g)$.

The simple continued fraction of a quadratic irrational is eventually periodic. For a nonsquare positive integer $D$, the continued fraction of $\sqrt{D}$ is periodic after its integer part.

For positive $a_j,b_j$, define $p_0=a_0$, $p_{-1}=1$, $q_0=1$, $q_{-1}=0$ andThenand $p_n/q_n$ is the corresponding finite generalized continued fraction.

The $k$th continued fraction convergent of $[a_0;a_1,a_2,\dots ]$ is its truncation $p_k/q_k=[a_0;\dots ,a_k]$, written in lowest terms with positive denominator. The recurrences are $p_k=a_k{p}_{k-1}+p_{k-2}$ and $q_k=a_k{q}_{k-1}+q_{k-2}$, initialized by $(p_{-2},p_{-1})=(0,1)$ and $(q_{-2},q_{-1})=(1,0)$.

For $\sqrt{2}=[1;2,2,\dots ]$, the convergents begin $1,3/2,7/5,17/12$. For an infinite simple continued fraction, $|\alpha -p_k/q_k|<1/(q_k{q}_{k+1})$. Thus the successive convergent denominators quantify the improving accuracy rather than merely recording a symbolic truncation.

Starting from $x_0=x>0$, definewhenever $x_n$ is not an integer. This produces $x=[a_0;a_1,a_2,\dots ]$.

A context-free grammar can enforce several matched counts by nesting their dependencies. For example,generates the context-free language $\{0^n{1}^m{0}^{n+m}:n,m\ge 0\}$. Applying the first recursive production rule $n$ times places $n$ zeros at both ends. Applying the second recursive production rule $m$ times places $m$ ones before the inner block and $m$ further zeros after it. ThusEvery terminal grammar derivation has this form, because it must switch from $S$ to $T$ exactly once. The nesting is useful: the final block records the sum of two independently chosen counts without requiring a context-sensitive grammar.

The productions $S\to 0S0\mid 1S1\mid 00\mid 11$ generate exactly the nonempty binary palindromes of even length.

A context-free grammar consists of terminal and nonterminal alphabets, a start symbol, and productions whose left side is one nonterminal. A derivation records the order of substitutions, while a parse tree records their hierarchical structure, so several derivations can represent one parse tree.

The real cyclotomic fieldis Galois of degree eight with cyclic Galois group $(\mathbb{Z}/17\mathbb{Z}{)}^{\times }/\{\pm 1\}$. A subgroup chain of successive index two gives a tower of three quadratic extensions, proving that $\cos (2\pi /17)$ is constructible.