Five-Value Theorem of Nevanlinna

In German known as Fünf-Punkte-Satz. This theorem is astounding. It says: If two meromorphic functions share five values ignoring multiplicity, then both functions are equal. Two functions, f(z) and g(z), are said to share the value a if f(z) - a = 0 and g(z) - a = 0 have the same solutions (zeros).

More precisely, suppose f(z) and g(z) are meromorphic functions and a_1, a_2, \ldots, a_5 are five distinct values. If

\displaystyle{      E(a_i,f) = E(a_i,g), \qquad 1\le i\le 5,  }


\displaystyle{      E(a,h) = \left\{ z | h(z) = a \right\},  }

then f(z) \equiv g(z).

For a generalization see Some generalizations of Nevanlinna’s five-value theorem. Above statement has been reproduced from this paper.

The identity theorem makes assumption on values in the codomain and concludes that the functions are identical. The five-value theorem makes assumptions on values in the domain of the functions in question.

Taking e^z and e^{-z} as examples, one sees that these two meromorphic functions share the four values a_1=0, a_2=1, a_3=-1, a_4=\infty but are not equal. So sharing four values is not enough.

There is also a four-value theorem of Nevanlinna. If two meromorphic functions, f(z) and g(z), share four values counting multiplicities, then f(z) is a Möbius transformation of g(z).

According Frank and Hua: We simply say “2 CM + 2 IM implies 4 CM”. So far it is still not known whether “1 CM + 3 IM implies 4 CM”; CM meaning counting multiplicities, IM meaning ignoring multiplicities.

For a full proof there are books which are unfortunately paywall protected, e.g.,

  1. Gerhard Jank, Lutz Volkmann: Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen
  2. Lee A. Rubel, James Colliander: Entire and Meromorphic Functions
  3. Chung-Chun Yang, Hong-Xun Yi: Uniqueness Theory of Meromorphic Functions, five-value theorem proved in §3

For an introduction to complex analysis, see for example Terry Tao:

  1. 246A, Notes 0: the complex numbers
  2. 246A, Notes 1: complex differentiation
  3. 246A, Notes 2: complex integration
  4. Math 246A, Notes 3: Cauchy’s theorem and its consequences
  5. Math 246A, Notes 4: singularities of holomorphic functions
  6. 246A, Notes 5: conformal mapping, covers Picard’s great theorem
  7. 254A, Supplement 2: A little bit of complex and Fourier analysis, proves Poisson-Jensen formula for the logarithm of a meromorphic function in relation to its zeros within a disk

Embedded Documents in MS Word

Recently I had trouble opening PDF documents embedded in Microsoft Word. I speculated about the following reasons for this problem:

  1. Adobe Acrobat not installed, I use MuPDF
  2. Embedded document simply not “really” embedded but just an icon

There is an easy solution. Microsoft Word 2007 documents are just zip-files. Just look at output of

7z l WordDocument.docx

The folder word/embeddings contains the embedded files, regardless of its type.

By the way, LibreOffice/OpenOffice files are also just zip-files.

Youtube 500 Internal Server Error

As noted in Youtube 500 Internal Server Error today I again noted an “500 Internal Server Error”. Normally you would not expect these kind of errors from Google. It says:

Sorry, something went wrong.

A team of highly trained monkeys has been dispatched to deal with this situation.

If you see them, send them this information as text (screenshots frighten them):


Converting WordPress Export File to Hugo

I have written on the Hugo static site generator here. Now I have written a migration program in the Go programming language to convert from WordPress export format to Hugo format. This program wp2hugo.go is in GitHub. It can be freely downloaded and does not need any further dependencies, except, of course, Go. The Go software is in Arch Linux or Ubuntu.

To convert a blog from WordPress you have to create an export file.

If the blog is not too voluminous one downloads a single XML-file which contains all posts and pages. If the blog in question is larger then you will receive an e-mail from that you can download a ZIP which contains two or more XML files in them. If you have such a ZIP-file, then unpack it, for example by using p7zip. Then run

go run wp2hugo.go XML1 XML2 ...

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Using Odroid as IP Router

I purchased an Odroid-XU4 for ca. 80 EUR including power-supply and case from Pollin. The original manufacturer is hardkernel. I intended to use this small ARM computer as a router and firewall. In the past I had used routers from multiple vendors, e.g., Linksys/Cisco, TP-Link, AVM/FritzBox, Netgear, and so on. There is a rule of thumb with all these devices: Usually you have to reboot them once or twice a month, otherwise they misbehave somehow. At least three of these device went completely catatonic. Now I had enough of this, I also wanted a command line interface to the router, ideally a real Linux system with bash, cron, gcc, etc. Although I already own an Intel NUC and I am very happy with this computer, an Intel NUC is a little bit too expensive to be used as just a router.

I recommend to additionally purchase a RTC backup battery. The Odroid has a realtime clock, but loses all date and time information once powered off. This way the log of the computer is garbled.

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