At the end of my article Hyperelliptic action integral, Annales de l'institut Fourier 49(1), p.303–331, I state the following conjecture:

Conjecture about a singularity
Let D be the open unit disk in the complex plane and be an open cover of the puntured disk D*= D\{0}. Suppose on each open set there is an injective holomorphic function such that on every intersection . Then those differentials glue together to a meromorphic 1-form on D.

It is evident that the 1-form is holomorphic on D*. In the case that its residue at the origin vanishes, the conjecture follows because the singularity can be at most a pole, as can easily be derived from Picard' big theorem (cited below). But I am unable to prove the conjecture for the case of a non-zero residue.
I would be grateful for any proof or counter-example — well, trule spoken, a counter-example would be less welcome since I hope that the conjecture is true. I am guided by my simple geometric intuition on Riemann surfaces...

In 1880 Charles Emile Picard (1856-1941) proved the following theorem.

Picard's Big Theorem
A holomorphic function takes in any neighborhood of an essential singularity every complex value, with at most one exception, infinitely often.

A typical example for Picard's theorem

The function defined by

is holomorphic on and has an essential singularity in . Is there a value that is does not take (Picard says "at most one exception")? Yes, since for all , there is an exceptional value and it must be zero; according to Picard's theorem there are, for any complex value and any , infinitely many complex numbers such that and .

Direct compuation of this example

In fact, for the example above the theorem of Picard is not really necessary, because we can also see what happens by direct computation: let be a non-zero complex value and let There existe two real numbers and such that

For let and Thus we have
Hence

By taking big enough we see that has infinitely many pre-images in the punctured disk

A less simple example

Denote by P the set of all prime numbers and consider the following function
 
Since there is an essential singularity we can apply Picard's Theorem.
But a direct computation seems impossible...

Short musical interlude — or rather a machine interlude since these guys look like tin soldiers and play the most complex rhyhmical patterns with the precision of a swiss watch. The drum group Top Secret Drum Corps from Basel is extremely precise, on a local and on a global level.

Just for comparison a video of the brazilian Batucada or Bateria Batala, where I played the Caixa (snare drum) for some years. We don't have the same precision as the swiss but nevertheless a lot of fun ;-) Viva la Samba in Paris!

At the beginning of the video clip there ere only the repiniques and the caixas, the girls in the first row start join in later with another rhythm on their big drums called surdos.

Classification

• There are three kind of people: those who know how to count and those who don't.
• There are two classes of people: those who think that mankind can be divided in two classes and those who doubt it.
• There are 10 sorts of people: those who understand the binary notation and those who don't.

How many mathematicians do you need to change a light bulp?
None. Mathematicians can't change light bulps but they can prove that it is feasable.

How many researchers in analysis do you need to change a light bulp?
Three. One for the existence, one for the uniqueness and one to determine the initial conditions.

How many researchers in numerical analysis are necessary to change a light bulp?
3.9967 (after six itérations).

How many constructivists are necessary for the change a light bulp?
Impossible. They don't belive in infinitesimal rotations.

How does a bourbakist change a light bulp?
Light bulp changing is a special case of the more general problem concerning the maintenance of an electric system. In order to establish a lower and an upper bound of the number of persons necessary for such a task, we must check whether the conditions of lemma 2.1 (disponibility of the staff) and the conditions of corollarys 2.3.55 (staff's motivation) are fulfilled. Only in the case where all those conditions are fulfilled we can obtain the result by applying the theorem from section 3.11.23. We recall that the upper bound is to be taken in a measure space equipped with the weak-*-topology.

One should think that elementary school taught us all that is to be known about basic operations on numbers: adding, substracting, multiplying and dviding. But there are still some surpises left! Here are three distinct ways to compute the product of two integers.

• The classical multiplication of the perfect pupil





• The lazy pupil's method (or Ancient egyptian multiplication)
 
The lazy man masters multiplication and division only by 2.



Instruction: In the left column take the half in each step; round-down if necessary. In the right column always double from one line to the next. Then erase all lines that begin with an even number (black) and add the remains on the right side (red).
 
 

• Method of Karatsuba (published in 1962)

Separate each factor on two halfs

and compute the following partial products:

The final result is obtained like this:

Remark
The up-shot of all this is the reduction of our computation to the products of one-digit numbers. On a computer the choice of an efficient algorithm can reduce considerably the time necessary to perform the program — when numbers with several milliards of digits are involved the saved time can be of the order of some days! Computing with such big numbers is not only interesting from a theoretical viewpoint, but has practical applications such as in cryptologie.
 
Exercices

1. Why does the lazy man's method work? Both factors play different roles; when you are given two numbers what casting do you propose?
2. With the help of the Karatsuba's multiplication compute the product 3116 x 1014. Explain in general, why the Karatsuba method works.
3. When you use the classical multiplication of the perfect pupil to compute the product of two n-digit integers, how many multiplications of one-digit numbers do you need to do?
4. By re-iteration of Karatsuba's method (separating the factors into, two, fours, eight, etc. parts) one constructs an slgorithm, the socalled Karatsuba-Algorithm. >When you apply it to the product of two n-digit intergers, how many single-digit products do you need? Compare the classical algorithm with the one of Karatsuba and show in particular, that the one of Karatsuba is much faster, being of the order .

Solutions
Here are the answers to this exercice in pdf-format.

Last but not least, a video about another multiplication method which produces a beautiful calligraphy — therefore we call it chinese multiplication!

The basic idea of the so-called chinese multiplication is this: a set of n parallel lines intersects another set of m parallel lines in nxm points. So there is nothing mysterious about that so-called chinese multiplication. Note that in the examples they always choose numbers with small digits like 112 or 2131 because with numbers like 89 or 897 there would be too many lines to draw!