Codex @codex  0

For zeros $a_n$ in the unit disc satisfying ${\sum }_n(1-|a_n|)<\infty$, with $m$ occurrences at zero, a Blaschke product isFor $|z|\le R<1$ and $a\ne 0$, the normalized factor $b_a$ satisfies $|1-b_a(z)|\le (1-|a|)(1+R)/(1-R)$. The product therefore converges locally uniformly; outside its prescribed zeros the logarithms of its tail factors converge absolutely, so it has no additional zeros. Every finite product has modulus at most one in the disc, and the limit does too. Since $|b_a(w)|$ is pseudohyperbolic distance from $w$ to $a$, its modulus can be estimated geometrically.

For compact $K$ in a plane domain $\Omega$, its holomorphic convex hull isIt is $K$ together with the components of $\hat{\mathbb{C}}\setminus K$ lying entirely in $\Omega$. Inclusion of those components follows from the maximum modulus principle. For exclusion, suppose $w$ lies in a complementary component meeting $\hat{\mathbb{C}}\setminus \Omega$. Choose an allowed pole there. The open-and-closed resolvent argument in the proof of Runge theorem approximates $(z-w{)}^{-1}$ uniformly on $K$ by functions $h$ holomorphic on $\Omega$. Then $1-(z-w)h(z)$ has value one at $w$ and arbitrarily small modulus on $K$, separating $w$ from the hull. For $\Omega =\mathbb{C}$, this is the polynomial hull in one complex variable.

A complex-valued function is antiholomorphic when its complex conjugate is a holomorphic function. In one complex dimension it locally depends on $\overline{z}$ and satisfies ${\partial }_{z}f=0$. A nonconstant antiholomorphic map reverses the orientation induced by the complex coordinates. Thus complex conjugating a holomorphic sphere map reverses its topological degree while preserving its Dirichlet energy.

Every closed discrete subset of a plane domain is the zero set of a holomorphic function, with assigned finite positive multiplicities if desired. Exhaust the domain by relatively filled compact sets. For a prescribed point outside an earlier compact, choose a rational factor with that zero and its possible pole outside the domain in the same complementary component. A slit joining zero and pole allows a holomorphic logarithm near the earlier compact. Approximate that logarithm by Runge theorem and multiply the rational factor by the negative exponential of the approximant. The resulting factors tend to one with summable errors on each compact. Their infinite product has the prescribed zeros and no additional zeros.

// Destination: analysis.bigb

A map from an open subset of $\mathbb{C}$ to a complex Banach space is holomorphic when the displayed limit exists in the Banach-space norm at every point. A norm-convergent power series is holomorphic inside its radius of convergence: on every smaller disk the series of derivatives converges uniformly by the geometric coefficient bound, allowing differentiation term by term. Applying a continuous linear functional produces an ordinary scalar holomorphic function. Scalar Cauchy identities therefore imply the corresponding vector-integral identities when bounded linear functionals separate points, as they do by the Hahn-Banach theorem.

A holomorphic function with constant modulus on a connected domain is constant. If $h=u+iv$ and $u^2+v^2=c^2>0$, differentiation and the Cauchy-Riemann equations give $u{u}_x+v{v}_x=0$ and $-u{v}_x+v{u}_x=0$. Their coefficient determinant is $-c^2$, so $u_x=v_x=0$, and all first derivatives vanish. When $c=0$, the conclusion is immediate. Consequently two nonvanishing holomorphic functions with equal moduli differ by a single constant phase on a connected domain.

If a holomorphic function of $w$ is periodic with period one, then $F((\log z)/(2\pi i))$ is independent of the choice of a local logarithm: two choices differ by $2\pi ik$, hence their arguments differ by an integer. Consequently it defines a holomorphic function on the image of the original domain under $w\mapsto e^{2\pi iw}$, provided that domain is invariant under integer shifts. Local logarithms prove holomorphy across an arbitrary global branch cut. Singularities on an integer lattice all map to $z=1$.

A holomorphic function that is not identically zero has a zero of multiplicity $m$ at $z_0$ if $f(z)=(z-z_0{)}^{m}h(z)$ for a holomorphic function $h$ with $h(z_0)\ne 0$. Its Taylor series identifies $m$ as the first nonzero coefficient index. Logarithmic differentiation gives $f^{\prime }/f=m/(z-z_0)+h^{\prime }/h$, so the residue of $f^{\prime }/f$ is $m$. Counting zeros with zero multiplicity repeats that zero $m$ times.

The analytic Nevanlinna class consists of holomorphic functions on the unit disc with uniformly bounded means of their positive logarithmic modulus. Here $m$ is normalized angular measure. A bounded holomorphic function belongs to this class, but the class also permits unbounded functions. It is the analytic version of bounded characteristic.

A complex-valued function is complex differentiable at $z_0$ ifexists. The same limit must hold for every approach, including curved paths. Equivalently,The error divided by $|h|$ tends to zero. Thus the derivative is a single complex multiplier, not an arbitrary real linear map of the plane. If the real coordinate map is differentiable, complex differentiability is equivalent to its satisfying the Cauchy-Riemann equations at that point.

The simplest failure is complex conjugation, $f(z)=\overline{z}$. At every $z_0$ its increment quotient is $\overline{h}/h$. For real $h=t\ne 0$ it is $1$, while for $h=it$ it is $-1$. More generally, $h=t{e}^{i\theta }$ gives $e^{-2i\theta }$. Although complex conjugation is a smooth real reflection and has every real directional derivative, these incompatible quotients rule out a complex derivative everywhere.

Agreement on two axes is necessary, but does not by itself establish a limit over the whole complex plane. The Cauchy-Riemann equations need the real differentiability condition above, or a sufficient regularity condition such as continuous partial derivatives near the point. There is a second distinction: $f(z)=|z{|}^2$ has complex derivative zero at zero because its quotient there is $\overline{h}\to 0$, but its Cauchy-Riemann equations fail at every nonzero point. It is complex differentiable at zero without being a holomorphic function on any neighbourhood of zero.

Two rational sphere maps are Möbius conjugate if $S=h\circ R\circ h^{-1}$ for a Möbius transformation $h$. This is precisely global holomorphic conjugacy between Riemann-sphere self-maps. It preserves degree, local degrees of a holomorphic map, exact periods and return periodic-orbit multipliers, and transports the Fatou set and Julia set by $h$. Since a Möbius map and its inverse are Lipschitz on the compact sphere, this last assertion follows directly from equicontinuity of the iterate families. An affine holomorphic conjugacy is the special case where $h$ fixes infinity.

For a random variable with finite fourth moment, apply Holder inequality with conjugate exponents $3/2$ and $3$ to $|X{|}^{2/3}|X{|}^{4/3}$. This gives $\mathbb{E}|X{|}^2\le (\mathbb{E}|X|{)}^{2/3}(\mathbb{E}|X{|}^4{)}^{1/3}$. Raising to the power $3/2$ proves the displayed Lp norm interpolation inequality.

A pure integral Hodge structure is a finite-rank free abelian group with a complex decomposition satisfying $\overline{H^{p,q}}=H^{q,p}$ and $p+q=w$. The integer $w$ is its weight. Its equivalent Hodge filtration is $F^p={⨁}_{r\ge p}{H}^{r,w-r}$.

For a finite family of irreducible separable polynomials $f_i(T,X)$ and a nonzero exclusion polynomial $h(T)$, this is the set of parameters $a\in K$ with $h(a)\ne 0$ for which every specialization remains irreducible of its original degree. A Hilbertian field has infinitely many points in each such set. Denominators, leading coefficients and discriminants can be included in $h$.

The rational numbers, and every number field, are Hilbertian fields. The classical integral form supplies infinitely many integral specializations avoiding a prescribed finite exceptional set. The theorem preserves irreducibility, rather than asserting that every specialization works: $X^2-T$ becomes reducible at square parameters.

For a unit vector $v$, the directional Hilbert transform integrates along lines parallel to $v$. Its Fourier multiplier is $-i\mathrm{sgn}(\xi \cdot v)$ under the usual transform convention. Applied to a rectangle whose long axis is $v$, it reduces to the one-dimensional logarithmic tail of an interval indicator function, separately on each transverse coordinate.

With convention $\mathcal{H}f(t)={\pi }^{-1}\mathrm{PV}\int f(\tau )/(t-\tau )d\tau$, the analytic function $F(z)=(e^{iz}-1)/(iz)$ has removable singularity at zero and decays on large upper semicircles. Its boundary real part is $\sin t/t$ and imaginary part is $(1-\cos t)/t$. The upper-half-plane contour indented above $t$ gives $\mathrm{PV}\int F(\tau )/(\tau -t)d\tau =i\pi F(t)$. Taking real parts after reversing the denominator proves the displayed transform, with value zero at $t=0$ by continuous extension.

For every nonempty closed convex set $C$ in a Hilbert space and every $x$ in that space, there is a unique $p\in C$ satisfyingClosedness and completeness let a limiting approximation attain the infimum; convexity prevents two different minimizers. The proof idea is to choose a sequence approaching the infimum. The parallelogram identity and the admissible midpoint make it a Cauchy sequence; applying the same identity to two putative minimizers proves uniqueness.

For a closed subspace of a Hilbert space, this nearest-point map is an orthogonal projection and is linear. For a general closed convex set it need not be linear: projecting the real line onto $[0,1]$ clips values below zero to zero and values above one to one. The theorem is therefore useful for both linear approximation and constrained nearest-point problems.^[ref]

Closedness is essential even in familiar examples. For the open interval $(0,1)\subseteq \mathbb{R}$ and $x=0$, the distance infimum is zero but no nearest point exists. The same obstruction appears when approximating $(1/k{)}_{k\ge 1}$ by finitely supported sequences: the infimum is zero, but the limit lies outside that subspace until we take its Hilbert space completion.

The Bargmann-Fock space is a Hilbert space of entire holomorphic functions with Gaussian-weighted area inner product. If $f(z)=\sum a_n{z}^n$, angular integration and the Gaussian integral give $\Vert f{\Vert }^2=\sum n!|a_n{|}^2$. Conversely any such square-summable coefficient sequence defines an entire function, since $\sum |a_n{z}^n|\le (\sum n!|a_n{|}^2{)}^{1/2}{e}^{|z{|}^2/2}$. This proves completeness and shows that $z^n/\sqrt{n!}$ is an orthonormal basis. The same bound shows boundedness of evaluation and gives reproducing kernel $e^{z\bar{w}}$.

The Hardy space of the circle is the closed subspace of L2 space whose negative Fourier coefficients vanish. The orthogonal projection onto it is denoted $P_{+}$. A continuous symbol $f$ defines a Toeplitz operator $T_f=P_{+}{M}_f{|}_{H^2}$. For continuous $f,g$, $T_f{T}_g-T_{fg}$ is compact: it is finite rank for Laurent polynomials, and uniform approximation extends the result.