Hurwitz Polynomial Puns

When we are given a complex polynomial, say F(s), to check whether the given polynomial is Hurwitz or not, we separate the given polynomial into odd and even powered polynomials For example, if,

F(s)= s^5 + 5s^4 +2s^3 + 3s^2 +8s+7

M(s)= s^5 + +2s^3 +8s

N(s)= 5s^4 +3s^2 +7

Then we check whether F(s) is Hurwitz or not by continued expansion of M(s)/N(s) and look at the sign of the coefficients of the quotients we obtain during the process, If all of them are positive, we conclude that F(s) is Hurwitz.

My question is; Why does this work? I've got some understanding as to why the continued fraction expansion of a fraction like 159/18 works, but how does this work for a single polynomial? What happens when we separate it into two polynomials; odd and even?

The problem that's stumping me is from Needham's Visual Complex Analysis; this is how it's stated in the text.

If Γ denotes the contour the hint describes moving with positive (anticlockwise, by the convention in the book) sense and the semi-circular section in the left-half plane, then I can see that if all n roots of F(s) lie in the left-half plane then Γ would enclose them all as R → ∞.

The argument principle then gives me that the winding number about the origin of the image of Γ under F would be n: v[F(Γ),0] = n.

Translating this observation into F(s) rotating through nπ as s moves from -i∞ to +i∞ is giving me some trouble, though. I tried plotting out F(Γ) in numpy for a few Hurwitz (and non-Hurwitz) polynomials to try and get a feel for what rotation happened along the straight-line segment, but it didn't seem to help: either my understanding of the problem or the script itself was flawed, since the segment seemed to map to a straight line (which would correspond to a net rotation of 0 or π/2 depending on whether it lay along the real or imaginary axis?).

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

Do your worst!

I'm surprised it hasn't decade.

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

It really does, I swear!

They’re on standbi

Pilot on me!!

Nothing, he was gladiator.

Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.

This sub is mostly just NSFW puns now.

If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.

Edit* I'm not replying any longer and turning off notifications but to all those that say "no one cares", there sure are a lot of you arguing about it. Maybe I'm wrong but you people don't need to be rude about it. If you really don't care, don't comment.

When I got home, they were still there.

What did 0 say to 8 ?

" Nice Belt "

So What did 3 say to 8 ?

" Hey, you two stop making out "

I won't be doing that today!

[Removed]

This morning, my 4 year old daughter.

Daughter: I'm hungry

Me: nerves building, smile widening

Me: Hi hungry, I'm dad.

She had no idea what was going on but I finally did it.

Thank you all for listening.

You take away their little brooms

There hasn't been a post all year!

Why

It’s pronounced “Noel.”

After all his first name is No-vac

What, then, is Chinese rap?

Edit:

Notable mentions from the comments:

• Spanish/Swedish/Swiss/Serbian hits

• French/Finnish art

• Country/Canadian rap

• Chinese/Country/Canadian rock

• Turkish/Tunisian/Taiwanese rap

There hasn't been a single post this year!

(Happy 2022 from New Zealand)

Nothing, it just waved

Him: I can explain everything!

(It's his best joke yet I think)

Bob

So that I could frequently say, "I am going to walk 5 miles now."

Edit: My most popular post on Reddit! 😀 Thank you for the awards.

Just to clarify, 12345678

Me grabbing a soda from my (what I thought was) half full 12pk...

Notices there's only 2;

Me: "Awe man... This is a damn bird box!" Her: "What the hell does that mean?!" Me: (Pulls both cans out & shows them to her) "It's only got Toucans."

I'm not ashamed to admit the look on her face was glorious.

I was just sitting there doing nothing.

But nobody ever mentions his sister, Onya, who invented the starter pistol.

“BOOM”?!

free