How many zeroes ? -B

[Camillo]

How many solutions has the equation $e^z+3z^2=0 $ in $|z| <1 $?

[Zero87]

How many solutions has the equation $e^z+3z^2=0 $ in $|z| <1 $?

Many months ago, my professor of complex analisys said me the following question:

"How many solutions has the equation $e^z+az^n =0$ in $|z|<1$, with the condition that $a\in \RR$ and $a>1$?"

That was during my exam of complex analisys, after she asked me the Rouché's Theorem...

For now, i won't to give a solution to the original question, i have already analized this other problem few moments ago!

post575081.html

Hi! See you later!

[Zero87]

How many solutions has the equation $e^z+3z^2=0 $ in $|z| <1 $?

I have corrected my proof in the other question too (post575081.html).

Testo nascosto, fai click qui per vederlo

So, in this equation, let $f(z)=3z^2$ and $g(z)=P(z)-f(z)=e^z$.
In $|z|=1$ we have $|f(z)|=3|z|^2=3$ and $|g(z)|=e^{Re(z)} cos(Re(z)) \le e^{Re(z)} \le e$. $^(1)$
So we can conclude, for Rouche's theorem, that $P(z)$ have the same number of zeroes of $f(z)$ in the domain $|z|<1$: $P(z)$ has 2 zeroes in the unit disk.

$^(1)$We have to give a motivation about the last inequality.
Let $z\in \CC$: we can write $z=Re(z)+i$ $Im(Z)$ in which $Re(z)$ is the real part of $z$ and $Im(z)$ is the imaginary part of the considered complex number.
We have $|z|^2=(Re(z))^2+(Im(z))^2$ so $(Re(z))^2=|z|^2-(Im(z))^2 \ge 0$ so $Re(z)= +- \sqrt{|z|^2-(Im(z))^2}$.
Now, if we take the positive solution, we have $e^(Re(z))\le e^|z| = e <3$; if we take the negative solution, we know that $e^x<1$ if $x<0$ so the result is correct in both cases.

[j18eos]

...i won't to give...

It's stronger than me: "io" in English is "I" and not "i"!

[Zero87]

It's more strong of me: "io" in English is "I" and not "i"!

Thank you.

[Paolo90]

It's more strong of me:

"... stronger than..."

[j18eos]

Thank you Paolo90, I thought it that there was a grammatical error!

[Roxie]

How many solutions has the equation $e^z+3z^2=0 $ in $|z| <1 $?

The equation has zero solutions, hasn't it?

[Zero87]

How many solutions has the equation $e^z+3z^2=0 $ in $|z| <1 $?

The equation has zero solutions, hasn't it?

The equation has no solutions, but in the Real line, not in the Complex field...

[Roxie]

Yeah, you're right... I hadn't thought about it, maybe because I haven't studied the complex field yet!!