- Suppose f ∈ℳ. Then f is a rational function.
- First note that as shown above, there are finitely many poles of f so f is analytic for> r
- Suppose f has a removable singularity at ∞. First of all, let S
_{i}be the singular part of the Laurent series expanded about α_{i}. Explain whyThen lim

_{z→0}zf_{n}= 0. Explain why f_{n}= ∑_{k=0}^{∞}a_{k}z^{k}for small z. Hence, for large,f_{n}= ∑_{k=0}^{∞}a_{k}and so f_{n}is bounded for large. Now explain why f_{n}is bounded and use Liouville’s theorem. - Next case is when f has a pole at ∞. Then f−∑
_{i=1}^{n}S_{i}≡ f_{n},f_{n}an entire function. If f has a pole at ∞, then f_{n}also has a pole at ∞ and so f_{n}= ∑_{k=1}^{r}b_{k}+ g, for small. where gis analytic. Thus for large,f_{n}−∑_{k=1}^{r}b_{k}z^{k}= gand since g is analytic, it follows that f_{n}−∑_{k=1}^{r}b_{k}z^{k}is bounded for large. Then explain why f−∑_{i=1}^{n}S_{i}−∑_{k=1}^{r}b_{k}z^{k}is a bounded entire function, hence a constant so fis a rational function. Note that this shows that any f which is in ℳhas a partial fractions expansion.

- First note that as shown above, there are finitely many poles of f
- Explain why any rational function is in ℳ. Thus, with the preceeding problem, ℳequals the rational functions.
- Suppose you have f ∈ℳ, not ℳ. Show that if f has finitely many zerosand poles, then there is an entire function gsuch that
where p

_{i}is the order of the pole at β_{i}and r_{i}is the order of the zero at α_{i}. Hint: First show f= ∏_{k=1}^{n}^{rk}hwhere his meromorphic but has no zeros. Then hhas the same poles with the same orders as f. Then h−∑_{i=1}^{m}S_{i}= lwhere lis entire and the S_{i}are the principal parts hcorresponding to β_{i}. Argue now thatwhere q

is entire and can’t have any zeros. Next use Problem 14. - Let w
_{1},w_{2},w_{3}be independent periods for a meromorphic function f. This means that if ∑_{i=1}^{3}a_{i}w_{i}= 0 for each a_{i}an integer, then each a_{i}= 0. Hint: At some point you may want to use Lemma 17.1.4.- Show that if a
_{i}is an integer, then ∑_{i=1}^{3}a_{i}w_{i}is also a period of f. - Let P
_{N}be periods of the form ∑_{i=1}^{3}a_{i}w_{i}for a_{i}an integer with≤ N. Show there are^{3}such periods. - Show P
_{N}⊆^{2}≡ Q_{N}. - Between the cubes of any two successive positive integers, there is the square of a positive
integer. Thus
^{3}< M^{2}<^{3}. Show this is so. It is easy to verify if you show that^{3∕2}− x^{3∕2}> 2 for all x ≥ 2 showing that there is an integer m between^{3∕2}and n^{3∕2}. Then squaring things, you get the result. - Partition Q
_{N}into M^{2}small squares. If Q is one of these, show its sides are no longer than - You have
^{3}points which are contained in M^{2}squares where M^{2}is smaller than^{2}. Explain why one of these squares must contain two different periods of P_{N}. - Suppose the two periods are ∑
_{i=1}^{3}a_{i}w_{i}and ∑_{i=1}^{3}â_{i}w_{i}, both in Q which has sides of length no more than C∕N^{1∕2}. Thus the distance between these two periods has length no more thanC∕. Explain why this shows that there is a sequence of periods of f which converges to 0. Explain why this requires f to be a constant.

This result, that there are at most two independent periods is due to Jacobi from around 1835. In fact, there are nonconstant functions which have two independent periods but they can’t be bounded.

- Show that if a
- Suppose you have f is analytic and has two independent periods. Show that f is a constant. Hint: Consider a parallelogram determined by the two periods and apply Liouville’s theorem. Functions having two independent periods which are analytic except for something called a pole are known as elliptic functions. Much can be said about these functions but not so much in this book.
- Suppose f is an entire function, analytic on ℂ, and that it has two periods w
_{1},w_{2}. That is f= fand f= f. Suppose also that the ratio of these two periods is not a real number so vectors, w_{1}and w_{2}are not parallel. Show, using Liouville’s theorem, that fequals a constant. Hint: Consider the parallelogram determined by the two vectors w_{1},w_{2}and tile ℂ with similar parallelograms. Elliptic functions are those which have two periods like this and are analytic except for poles. Roughly, these are points wherebecomes unbounded. Thus the only analytic elliptic functions are constants. - You can show that if r is a real irrational number the expressions of the form m + nr for m,n integers
are dense in ℝ. See my single variable advanced calculus book or modify the argument in Problem 4.
(Let P
_{N}be everything of the form m + nr where,≤ N. Thus there are^{2}such numbers contained in≡ I. Let M be an integer,^{2}< M <^{2}and partition I into M equal intervals. Now argue some interval has two of these numbers in P_{N}etc.) In particular,can be made as small as desired. Now suppose f is a non constant meromorphic function and it is periodic having periods w_{1},w_{2}where if, for m,n integers, mw_{1}+ nw_{2}= 0 then both m,n are zero. Show that w_{1}∕w_{2}cannot be real. This was also done by Jacobi. - Suppose you have a nonconstant meromorphic function f which has two periods w
_{1},w_{2}such that if mw_{1}+ nw_{2}= 0 for m,n integers, then m = n = 0. Let P_{a}be a parallelogram with lower left vertex at a and sides determined by w_{1}and w_{2}such that no pole of f is on any of the sides. Show that the sum of the residues of f found inside P_{a}must be zero. - The modular group is the set of fractional linear transformations, such that a,b,c,d are integers and ad − bc = 1. Show this modular group is really a group with the group operation being composition. Also show the inverse ofis.

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