Math Help - complex variable

1. complex variable

Hey, I would really appreciate some hints on how to tackle this please:

For a natural number N, let CN be the square with vertices at ±(N + 1
2 )(1 ± i).
Show that there is a constant K so that
maxzeCN | cot(pi*z)| is less than or equal to K.

I know it could help to estimate the value of | cot(pi*z)| separately on each side of the square separately using the fact that

Thanks.

2. Originally Posted by dazed
Hey, I would really appreciate some hints on how to tackle this please:

For a natural number N, let CN be the square with vertices at ±(N + 1
2 )(1 ± i).
Show that there is a constant K so that
maxzeCN | cot(pi*z)| is less than or equal to K.

I know it could help to estimate the value of | cot(pi*z)| separately on each side of the square separately using the fact that

Thanks.
For $|x|\leq n + \tfrac{1}{2}$ we have that if $\alpha = x \pm i (n + \tfrac{1}{2})$ is on the horizontal sides. If taken with $+$ then it is the top and if taken with $-$ then it is the bottom.

This means, $\left| \cot \pi \alpha \right| = \left| \frac{\cos \pi \alpha}{\sin \pi \alpha} \right| = \left| \frac{e^{\pi i \alpha} + e^{- \pi i \alpha}}{ e^{\pi i\alpha} - e^{-\pi i \alpha} } \right| \leq \frac{| e^{\pi i \alpha} | + |e^{-\pi i \alpha} | }{ ||e^{\pi i \alpha}| - |e^{-\pi i \alpha}||} = \frac{e^{\pi(n+1/2)} + e^{\pi(n-1/2)}}{e^{\pi(n+1/2)} - e^{\pi (n-1/2)}} =$ $\coth \pi (n + \tfrac{1}{2})$

The function $f(x) = \coth \pi x,x>0$ is a decreasing function therefore, $\coth \pi (n+\tfrac{1}{2}) \leq \coth \tfrac{3\pi}{2}$.
This means that $\left| \cot \pi \alpha \right| \leq \coth \tfrac{3\pi}{2}$ for $\alpha = x \pm i (n + \tfrac{1}{2})$ where $|x| \leq n+\tfrac{1}{2}$

Now if $\beta = \pm (n+\tfrac{1}{2}) + i x$ with $+$ being the right vertical side and $-$ being the left vertical side then,
$|\cot \pi \beta| = | \cot \pi (n + \tfrac{1}{2} + ix) | = | \cot (\pi n + \tfrac{\pi}{2} + i\pi x) | =$ $|\cot ( \tfrac{\pi}{2} + i\pi x )| = |-\tan (i\pi x)| = |\tanh \pi x|$
However, by property of hyperbolic tangent we have $|\tanh \pi x| \leq 1$.

Therefore, $\cot \pi z$ is bounded on $C_n$ by the constant $\max \{ 1, \coth \tfrac{3\pi}{2} \}$.

3. Thanks so much for your help, though i'm still alil confused.

Doesn't cot(pi*z) = cos(pi*z)/sin(pi*z)?

x

4. Originally Posted by dazed
Thanks so much for your help, though i'm still alil confused.

Doesn't cot(pi*z) = cos(pi*z)/sin(pi*z)?

x
It does! But the proof still works out because I accidently written them the other way around.
It is fixed now.

5. Is there an i missing as well, so that:

cot(pi*z)=i(exp(i*pi*z)+exp(-pi*i*z))/exp(pi*i*z)-exp(-pi*i*z)?

x

6. Originally Posted by dazed
Is there an i missing as well, so that:

cot(pi*z)=i(exp(i*pi*z)+exp(-pi*i*z))/exp(pi*i*z)-exp(-pi*i*z)?

x
The equation is put into absolute value terms and $| i | = 1$.