I am attempting to learn an exercise in the chapter on Rouche's Theorem in Ahlfors Complex Analysis. It comes from page 154, number 3:   Find the number of zeros of  $$f(z) = z^4 + 8z^3 + 3z^2 + 8z + 3 = 0$$ in the right half plane.   My first thought was to try to use Rouche's Theorem, but this has been to no avail.  Does anyone have any thoughts as to how to approach this problem. I am almost sure that I am going to need to use a semi circle (in the right half plane) centered at zero. The question is how.
This is a question in Ahlfors in the section on the argument principle: How many roots of the equation $f(z)=z^4+8z^3+3z^2+8z+3=0$ lie in the right half plane?  He gives a hint that we should "sketch the image of the imaginary axis and apply the argument principle to a large half disk."  Since $f$ is an entire function, I think I understand that the argument principle tells us that for any closed curve $\gamma$ in $\mathbb{C}$, the winding number of $f(\gamma)$ around 0 is equal to the number of zeros of $f$ contained inside $\gamma$.  How would you go about actually applying the hint though? I am having trouble figuring out what the image of a large half disk under $f$ would look like.
Android - Call log/ history : Retrieve a dialed number when there was no network coverage I dialed a number when I had no network support but it does not appear in the call history of my android phone.  Is there any way to get it ? (system logs or whatever)  THanks a lot in advance, Jack I entered the USA on a three month visa waiver program from Australia. I am about to return to  Australia and would like to come back again to the USA. How long do I have to wait until I can re-enter the USA for another 3 months? Find all entire functions $f(z)$ such that $|f(z)|=1$ for $|z|=1$  Hint: First show that $f(z)$ is a polynomial.  Clearly one can not use Cauchy Estimates to prove that $f(z)$ is a polynomial, the other way is to prove that $f(z)$ has a pole at infinity which I am not sure how to prove that, are there any other ways to prove that a given entire function is a polynomial ?  Thank you ! When is the graph of the divisor relation on $[n]$ planar? When can we use 'much' in affirmative clauses? I found the below in  web-site:     We use much in questions and negative clauses to talk about degrees of something.      We don’t use much in affirmative clauses.   Also, it the same web-site, I found the following:     She runs much faster than he does.    also I noticed that some web-sites use much with affirmative clauses:     May we all be having this much fun and getting this much exercise when we're his age!    Is it Okey to use much in affirmative clauses? If yes, when can we use it? and what is the rules?  Update:  Here is another example on how often the word "much" is used  in informal style:     After much consideration I will NOT be performing at the Inauguration with my improv group Capitol Hill-arious Are global variables thread-safe in Flask? How do I share data between requests? What happens when a caster targets an object that looks like a creature with a spell that targets only creatures? There are many spells which explicitly target creatures:     Choose one creature within range   The target creature is normally supposed to make a saving throw in this case.  What happens when the caster chooses a creature, that is actually an object? For instance, a scarecrow, a mannequin, an illusion, or a corpse.  Examples:   An evil mage is hiding in the bushes. He casts an illusion of himself on the road, using the Silent Image spell. A PC sorcerer walks down the road, sees the illusion and casts Magic Missile on it. A cleric is standing near a pile of dead bodies. He has suspicions that a few corpses are actually undead creatures. He casts Sacred Flame on every corpse, see which if them succeed, hence, reveals the undead. A wizard casts Acid Splash on a mannequin, trying to damage it with acid.   Possible outcomes I can think of are:   The caster cannot even try to cast the spell, regardless of their information about what the target actually is. The caster tries to cast the spell, the casting fails, not expending the spell slot. The caster casts the spell, it expends the spell slot, then fails. The caster casts the spell, but it hits the original and not the illusion. The caster casts the spell, it flies to the target illusion but deals no damage to it (passes through it).   References in the rules and Sage advice  I was trying to investigate, but it didn't make things clearer:    has an answer, that Magic Missile won't hit. It says nothing about if it can be targeted at all, it leaves it up to the DM. Magic Missile automatically finds the original, even when the caster targets an illusion. At least . Now . This supports the counter-intuitive option 4.   on the other hand   , if there were no perceivable effects (let's say you can try to cast the Command spell on an illusion). That assumes that casters can waste their spells on illusion, which means they at least can target it. This supports options 3 and 5. 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