- Suppose f is an entire function with f= 1. Let
Use Jensen’s equation to establish the following inequality.

where n

is the number of zeros of f in B. - The version of the Blaschke product presented above is that found in most complex
variable texts. However, there is another one in [?]. Instead of you use
Prove a version of Theorem 56.5.1 using this modification.

- The Weierstrass approximation theorem holds for polynomials of n variables on any
compact subset of ℝ
^{n}. Give a multidimensional version of the Müntz-Szasz theorem which will generalize the Weierstrass approximation theorem for n dimensions. You might just pick a compact subset of ℝ^{n}in which all components are positive. You have to do something like this because otherwise, t^{λ}might not be defined. - Show cos= ∏
_{k=1}^{∞}. - Recall sin= zπ ∏
_{n=1}^{∞}. Use this to derive Wallis product,= ∏_{k=1}^{∞}. - The order of an entire function, f is defined as
If no such a exists, the function is said to be of infinite order. Show the order of an entire function is also equal to limsup

_{r→∞}where M≡ max. - Suppose Ω is a simply connected region and let f be meromorphic on Ω. Suppose
also that the set, S ≡has a limit point in Ω . Can you conclude f= c for all z ∈ Ω?
- This and the next collection of problems are dealing with the gamma function.
Show that
and therefore,

with the convergence uniform on compact sets.

- ↑ Show ∏
_{n=1}^{∞}e^{−z -n }converges to an analytic function on ℂ which has zeros only at the negative integers and that therefore,is a meromorphic function (Analytic except for poles) having simple poles at the negative integers.

- ↑Show there exists γ such that if
then Γ

= 1 . Thus Γ is a meromorphic function having simple poles at the negative integers. Hint: ∏_{n=1}^{∞}e^{−1∕n}= c = e^{γ}. - ↑Now show that
- ↑Justify the following argument leading to Gauss’s formula
- ↑ Verify from the Gauss formula above that Γ= Γz and that for n a nonnegative integer, Γ= n!.
- ↑ The usual definition of the gamma function for positive x is
Show

^{n}≤ e^{−t}for t ∈. Then showUse the first part to conclude that

Hint: To show

^{n}≤ e^{−t}for t ∈, verify this is equivalent to showing^{n}≤ e^{−nu}for u ∈. - ↑Show Γ= ∫
_{0}^{∞}e^{−t}t^{z−1}dt. whenever Rez > 0. Hint: You have already shown that this is true for positive real numbers. Verify this formula for Rez > 0 yields an analytic function. - ↑Show Γ=. Then find Γ.
- Show that ∫
_{−∞}^{∞}e^{−s2 -2- }ds =. Hint: Denote this integral by I and observe that I^{2}= ∫_{ℝ2}e^{−(x2+y2) ∕2 }dxdy. Then change variables to polar coordinates, x = r cos, y = r sinθ. - ↑ Now that you know what the gamma function is, consider in the formula for
Γthe following change of variables. t = α + α
^{1∕2}s. Then in terms of the new variable, s, the formula for ΓisShow the integrand converges to e

^{−}. Show that thenHint: You will need to obtain a dominating function for the integral so that you can use the dominated convergence theorem. You might try considering s ∈

first and consider something like e^{1−(s2∕4) }on this interval. Then look for another function for s >. This formula is known as Stirling’s formula. - This and the next several problems develop the zeta function and give a
relation between the zeta and the gamma function. Define for 0 < r < 2π
Show that I
_{r}is an entire function. The reason 0 < r < 2π is that this prevents e^{reiθ }− 1 from equaling zero. The above is just a precise description of the contour integral, ∫_{γ}dw where γ is the contour shown below.in which on the integrals along the real line, the argument is different in going from r to ∞ than it is in going from ∞ to r. Now I have not defined such contour integrals over contours which have infinite length and so have chosen to simply write out explicitly what is involved. You have to work with these integrals given above anyway but the contour integral just mentioned is the motivation for them. Hint: You may want to use convergence theorems from real analysis if it makes this more convenient but you might not have to.

- ↑In the context of Problem 19 define for small δ > 0
where γ

_{rδ}is shown below.Show that lim

_{δ→0}I_{rδ}= I_{r}. Hint: Use the dominated convergence theorem if it makes this go easier. This is not a hard problem if you use these theorems but you can probably do it without them with more work. - ↑ In the context of Problem 20 show that for r
_{1}< r, I_{rδ}−I_{r1δ}is a contour integral,where the oriented contour is shown below.

In this contour integral, w

^{z−1}denotes e^{}logwhere log= ln+ iargfor arg∈. Explain why this integral equals zero. From Problem 20 it follows that I_{r}= I_{r1}. Therefore, you can define an entire function, I≡ I_{r}for all r positive but sufficiently small. Hint: Remember the Cauchy integral formula for analytic functions defined on simply connected regions. You could argue there is a simply connected region containing γ_{r,r1,δ}. - ↑ In case Rez > 1, you can get an interesting formula for Iby taking the limit as r → 0. Recall that
Thus