MAT 542  Assignment No. 10  Due April 29, 2002

(1) This set of problems deals with the formula

(1)  sin πz = πz ∏_{n=1}^{∞} (1 − \frac{z^{2}}{n^{2}} ) .

(2) The product expansion of the sin-function was established in class as a consequence of the Weierstrass product theorem. It used the infinite sum formula for the cot-function:

(2)  π cot πz = \frac{1}{z} − 2z ∑_{n=1}^{∞} \frac{1}{n^{2} − z^{2}} .

(3) Below are two “other” proofs of formula (1). The first is a more formal consequence of (2). The second does not use (2) at all. As usual, supply justifications for all claims made.

(4) Rewrite (2) as

\frac{d}{dz} log sin πz = \frac{d}{dz} (log z + log ∏_{n=1}^{∞} (1 − \frac{z^{2}}{n^{2}} )) .

(5) Use the last equation to obtain (1).

(6) To start the second proof of (1), define the function

s(z) = πz ∏_{n=1}^{∞} (1 − \frac{z^{2}}{n^{2}} ) .

Show that the function s is entire, odd and has simple zeros at all the integers.

(7) Show that

s(z) = πz(1−z) ∏_{n=1}^{∞} (1 + \frac{z}{n}) (1 − \frac{z}{n + 1}) = πz(1−z) ∏_{n=1}^{∞} (1 + \frac{z(1 − z)}{n(n + 1)}) .

(8) Prove that s(1 − z) = s(z) and hence also s(z + 1) = −s(z). Define q(z) = \frac{s(z)}{sin πz} and conclude that q is entire and q(z + 1) = q(z).

(9) Show that

|∏_{n=1}^{∞} (1 − \frac{z^{2}}{n^{2}} )| ≤ e^{π|z|−2|z| cot^{−1} |z|},

2| sin πz| ≥ e^{πℑz} − e^{−πℑz} and that for some real constant A and B,

|q(z)| ≤ A + B|z| for |ℜz| ≤ \frac{1}{2}.

(10) Show that q(z) = 1.