Intereting Posts

Computing the Galois group of polynomials $x^n-a \in \mathbb{Q}$
If $F(a, b)=\langle a, B\rangle$ then $B=a^ib^{\epsilon}a^j$: a neat proof?
increasing subset of a partial order and characteristic function
What is the probability that this harmonic series with randomly chosen signs will converge?
Finding an equation for a circle given its center and a point through which it passes
What would be the Nth number?
Maximizing volume of a rectangular solid, given surface area
Let $S:U\rightarrow V, \ T:V\rightarrow W$ and if $S$ and $T$ are both injective/surjective, is $TS$ injective/surjective?
Group of order $60$
Repayments of a loan with compound interest
Axiom Systems and Formal Systems
Why there is no continuous argument function on $\mathbb{C}\setminus\{0\}$?
Infinite product of measurable spaces
Closed-form of an integral involving a Jacobi theta function, $ \int_0^{\infty} \frac{\theta_4^{n}\left(e^{-\pi x}\right)}{1+x^2} dx $
Bijection between the set of classes of positive definite quadratic forms and the set of classes of quadratic numbers in the upper half plane

A bit of a general question, but here goes. Morally, what is the space of germs of a holomorphic function?

I know that a germ is simply an equivalence class of function elements, where we regard two function elements as equivalent at a point if they agree on some open neighbourhood of that point. Moreover I know the definition that the space of germs is simply the union of these equivalence classes for all functions $f$ and points $x$.

This is all a bit abstract at the moment though. I can’t see how germs are useful, or how I might calculate them for a concrete function. Has anyone got any nice examples of calculations of the space of germs? And could someone explain the overarching idea behind them in Riemann Surfaces?

- How to solve an integral with a Gaussian Mixture denominator?
- “The Egg:” Bizarre behavior of the roots of a family of polynomials.
- If $\mathbb f$ is analytic and bounded on the unit disc with zeros $a_n$ then $\sum_{n=1}^\infty \left(1-\lvert a_n\rvert\right) \lt \infty$
- Uniform convergence of infinite series
- Homeomorphism between the Unit Disc and Complex Plane
- Application of Liouville's Theorem

Many thanks.

- To compute $\frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2|^2 dz$ where $\mathcal{C}$ is the unit circle in $\mathbb{C}$
- Ahlfors “Prove the formula of Gauss”
- Radial Limits for Holomorphic Functions
- Why is $2\pi i \neq 0?$
- contour integration of logarithm
- Isolated singularities of the resolvent
- Determine the Winding Numbers of the Chinese Unicom Symbol
- “Unsolvable” Equations
- Local and global logarithms
- Fourier transform on $1/(x^2+a^2)$

Consider the set of all germs of holomorphic functions at a specific point $x$. If you know of stalks, then this is just $\mathcal{O}_x$. You can think of this set as a collection of all “possible functions” which are holomorphic at that point.

To be precise, you can think of it as the set of all power series which converge in a little neighborhood of $x$. So $\mathcal{O}_x$ is isomorphic to the ring $\mathbb{C}\{z-x\}$ of all convergent power series in $z-x$.

Of course, two functions define the same germ if their power series about $x$ are the same. This gives a method to calculate the germs of a function $f$ at points $x$.

Now the union of all germs of all functions is useful for instance because it allows for the construction of a maximal analytic continuation of a given holomorphic function:

For a Riemann Surface $X$, a point $x\in X$ and a function germ $f$ at $x$, you get a Riemann Surface $Y$ together with an unbranched holomorphic map $Y\rightarrow X$ and a holomorphic function $F$ on all $Y$, such that the germ of $F$ is in some natural way the same as that of $f$.

Now the space of germs allows for the construction of a “maximal $Y$”. You can think of it as the largest possible domain of definition for $f$. Because there may be different continuations of a representative of $f$ in different neighborhoods, you have to consider them all and combine them to get your larger space $Y$.

The actual construction and proofs can be read, for instance, in O. Forster’s “Lectures on Riemann Surfaces”.

- What are the surfaces of constant Gaussian curvature $K > 0$?
- Why does synthetic division work?
- How to prove error function $\mbox{erf}$ is entire (i.e., analytic everywhere)?
- Summation of natural number set with power of $m$
- N unlabelled balls in M labeled buckets
- Is $p\mapsto \|f\|_p$ continuous?
- Evaluate $\int_0^{\infty}\frac{e^x-1}{xe^x(e^x+1)}dx.$
- Formal definition of a random variable
- Order of matrices in $GL_2(\mathbb{Z})$
- Proof that $\sum\limits_{j,k=1}^N\frac{a_ja_k}{j+k}\ge0$
- Anti-curl operator
- An identity on Stirling number of the first and the second kind.
- How many numbers less than $x$ have a prime factor that is not $2$ or $3$
- Line with two origins is a manifold but not Hausdorff
- Reference for multivariable calculus