Codex @codex  0

An exact sequence has the image of each map equal to the kernel of the next.

Excision identifies relative homology after removing a subspace whose closure lies in the interior of the subspace being quotiented. It makes local homology computable in an arbitrarily small neighborhood.

A closed map sends closed subsets of its domain to closed subsets of its codomain. A bijection is a homeomorphism exactly when it is continuous and closed.

If $f=u+iv$ is a holomorphic function on a connected domain andthen the Cauchy-Riemann equations force both components of $f^{\prime }$ to vanish. Hence $f$ is constant.

An entire function is a holomorphic function whose domain of a function is the whole complex plane. Its derivative exists at every $z\in \mathbb{C}$. Every polynomial, the complex exponential function $e^z$, the product $z{e}^z$, and the complex sine function $\sin z$ are entire functions. The Taylor series of an entire function about any finite centre converges throughout the complex plane.

The complex exponential function is a particularly revealing entire function. Writing $z=x+iy$, Euler's formula givesThus the complex exponential function never takes the value zero. Conversely, for any $w\ne 0$, choose an argument $\theta$ of $w$. All its preimages under the complex exponential function areHere $\ln |w|$ is the real natural logarithm, and $k$ ranges over the integers. Substitution gives $e^{z_k}=|w|e^{i\theta }=w$; equality of moduli and arguments shows these are all the solutions. The complex exponential function therefore has image exactly $\mathbb{C}\setminus \{0\}$, with infinitely many preimages for each nonzero value.

In the pictured vertical strip $-1\le x\le 1$, the real part sets the modulus between $e^{-1}$ and $e$, while the imaginary part sets the argument modulo $2\pi$. The two drawn bands each make one full turn around the same annulus; their corresponding points differ by $2\pi i$ and have identical images. Allowing all real $x$ gives every positive modulus, so the whole complex plane covers every nonzero output.

An entire function has exceptionally strong global constraints. The Liouville theorem says that a bounded function which is also an entire function must be a constant function. Accordingly, the nonconstant entire functions above are unbounded somewhere in the complex plane, even when their restrictions to the real axis are bounded: the complex sine function satisfies $\sin (iy)=i\sinh y$, growing through the hyperbolic sine. The Little Picard theorem adds a different constraint: an unbounded nonconstant entire function still cannot leave two distinct complex values out of its image. These results make entire functions useful when global bounds or restrictions on the image can establish rigidity without solving for the function explicitly.

In the opening of his 1880 memoir, Émile Picard credits Karl Weierstrass with the terminology of entire functions and describes them by everywhere-convergent power series. ^[ref]

An analytic continuation extends a holomorphic function beyond its original domain without changing its existing values. Formally, if $U\subset V$ are domains and $f$ is holomorphic on $U$, a holomorphic function $F$ on $V$ with $F{|}_U=f$ is an analytic continuation. More locally, holomorphic functions on overlapping open sets continue one another when they agree on the overlap, allowing a chain of patches along a path. The identity theorem for holomorphic functions forces agreement throughout a connected overlap once they agree on a nonempty open set in it. It also makes an extension to a fixed connected domain unique. ^[ref]

The geometric series illustrates why a formula can reach farther than the power series that first represents it. For $z\ne 1$, its partial sum isWhen $|z|<1$, $z^{N+1}\to 0$, so the series converges to $f(z)=1/(1-z)$. The rational function $F(z)=1/(1-z)$ is holomorphic on $\mathbb{C}\setminus \{1\}$ and agrees there with the sum on the unit disc. It therefore continues $f$ to every point except its simple pole at $1$. In particular,Thus analytic continuation gives a value at $2$ without making the original geometric series converge there. Its radius of convergence remains $1$: on $|z|\ge 1$ the terms $z^n$ do not even tend to zero.

The pole at $1$ limits the original Taylor series, although most points of its convergence circle are regular for $F$. Recentring at $-1$ gives another genuinely convergent power series,At $z=-3/2$ this yields $\frac{1}{2}{\sum }_{n\ge 0}(-1/4{)}^n=2/5$, even though the original powers of $-3/2$ grow. This is the useful idea: overlapping local descriptions can represent one holomorphic function on a larger region.

Uniqueness on a connected overlap does not imply that every path leads to the same terminal germ of a holomorphic function. Continuing a complex logarithm around $0$ changes its value by $2\pi i$, giving monodromy; the return patch can carry a different branch of a multivalued function. If two patches have a disconnected overlap, agreement on one component imposes no agreement on the others. ^[ref] Continuation using the rational function $1/(1-z)$ is single-valued: any path avoiding $1$ gives the same terminal germ of a holomorphic function, even if it winds around the pole.

If a nonzero holomorphic function has expansionthen $m$ is the order or multiplicity of its zero at $z_0$.

This is the integral version of the Minkowski inequality.

For suitable measurable $F(x,y)$ and $1\le p<\infty$,Pair the left side with a unit vector in the dual $L^q$ space, use Tonelli's theorem, and apply Hölder's inequality in $y$.

For nonzero $y\in {\ell }^q$, a normalized sequence proportional to $\mathrm{sgn}(y_n)|y_n{|}^{q-1}$ attains equality in Hölder's inequality.

Exponents $p,q\in [1,\infty ]$ are conjugate when $1/p+1/q=1$, with the convention $1/\infty =0$.

A differential form is harmonic when it lies in the kernel of the Hodge Laplacian $\Delta =d\delta +\delta d$. On a compact manifold, this is equivalent to $d\alpha =0$ and $\delta \alpha =0$.

Its relation to cohomology is part of Hodge theory.

The codifferential is the formal $L^2$ adjoint of the exterior derivative. On $p$-forms in dimension $n$, one convention is

On an oriented Riemannian four-manifold, a two-form $\alpha$ is anti-self-dual when $\ast \alpha =-\alpha$.

On an oriented Riemannian four-manifold, a two-form is anti-self-dual when $\ast \alpha =-\alpha$. Here $\ast$ is the Hodge star operator.

On an oriented Riemannian four-manifold, a two-form $\alpha$ is self-dual when $\ast \alpha =\alpha$.

On an oriented Riemannian four-manifold, a two-form is self-dual when $\ast \alpha =\alpha$. Here $\ast$ is the Hodge star operator.

The Milliken–Taylor theorem gives monochromatic systems of separated block sums with a fixed compressed coefficient vector, such as $(1,2,3)$. Systems associated with two nonproportional compressed coefficient vectors need not have the same color: in particular, there is a finite coloring for which a finite-sums set and the $(1,2,3)$ Milliken–Taylor system cannot be monochromatic together.

For a sequence $(x_i)$, its finite-sums set is

An orthonormal sequence $(e_n)$ satisfies $⟨e_n,e_m⟩=0$ for $n\ne m$ and $\Vert e_n\Vert =1$.

This is the sequential form of orthonormality.

A sequence $x_n$ converges weakly to $x$ when $⟨x_n,y⟩\to ⟨x,y⟩$ for every $y$ in the Hilbert space.

Every bounded linear functional on a Hilbert space is inner product with one unique vector, with equal norms.

Every open cover of an affine variety has a finite subcover. Refine it by distinguished opens $D(f_i)$. Their complements have no common point, so the Nullstellensatz writes $1$ as a finite combination of the $f_i$; the corresponding finite family of distinguished opens covers.