The Factor Theorem

April 28, 2009

I was a little angry the other day to be assigned the following problem for homework:

Suppose a_o,\dots,a_n are in \mathbb{C}. Show that \displaystyle p(z) = a_nz^n+a_{n-1}z^{n-1}+\cdots+a_0 = \prod_{i=1}^n (z-z_i) for z_i (not necessarily distinct) in \mathbb{C}. Assume the fundamental theorem of algebra.

Why the anger? Because I don’t know algebra; I’m not taking algebra. I found proofs on the internet, but they all require algebra that I simply don’t know. So I resorted to the following:

Lemma

If p(z) is a polynomial  of degree n with p\!:\!\mathbb{C}\rightarrow \mathbb{C}, then the Taylor polynomial of p at a is equal to p(z).

The proof of this statement is obvious and can be found in a couple of ways: show that the two polynomials are equal up to order n and therefore equal; find the k^{th} derivative of p at a and use that to show that a_k is also the coefficient for the k^{th} term of the Taylor polynomial.

Theorem

Suppose a_o,\dots,a_n are in \mathbb{C}. Then\displaystyle p(z) = a_nz^n+a_{n-1}z^{n-1}+\cdots+a_0 = \prod_{i=1}^n (z-z_i) for z_i (not necessarily distinct) in \mathbb{C}.

Proof

Given p(z) = a_nz^n+a_{n-1}z^{n-1}+\cdots+a_0 then

\displaystyle p(z) = \sum_{k=0}^n\! \frac{p^{(k)}(a)}{k!}(z-a)^k by the lemma. That is,

\displaystyle p(z) = p(a) + \sum_{k=1}^n\! \frac{p^{(k)}(a)}{k!}(z-a)^k

\displaystyle p(z) = p(a) + (z-a) \sum_{k=1}^n\! \frac{p^{(k)}(a)}{k!}(z-a)^{k-1}

Suppose p(a) = 0. That is, suppose a = z_1 where z_i are the roots of p(z). We know z_1 must exist by the fundamental theorem of algebra.

p(z) = 0 + (z-a) \displaystyle\sum_{k=1}^n\! \frac{p^{(k)}(a)}{k!}(x-a)^{k-1} = q(z)

Clearly, \displaystyle \frac{p(z)}{z-a} = q(z)

We continue in the same way; take a root z_2 of q(z) and repeat the above by taking the Taylor polynomial of q at a = z_2. This will eventually reduce the equations to:

\displaystyle \frac{p(z)}{(z-z_1)(z-z_2)\cdots(z-z_n)} = \frac{q(z)}{(z-z_2)(z-z_3)\cdots(z-z_n)}

\displaystyle = \cdots = \frac{z-z_n}{z-z_n} = 1

Considering only the first and last terms, it is clear that

\displaystyle p(z) = (z-z_1)(z-z_2)\cdots(z-z_n) = \prod_{i=1}^n (z-z_i)

QED

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Posted by Jay

Ilosaarirock: First Impressions

April 9, 2009

ilosaarirockHaving spent a solid 3 hours now with Porcupine Tree’s Ilosaarirock, I feel confident calling myself an expert on it. If you don’t want to read my more thorough description of each song, here’s the TLDR version: Finland sucks, Porcupine Tree rocks.

  • Fear of a Blank Planet: Yes. This sounds just like it did when I heard them for their US FOABP tour. It’s everything I dreamed of.
  • Lightbulb Sun: Actually, wow. This is my favorite recording of the song; I like it even more than the Warszawa version. I find something really appealing about its hardcore-ness.
  • Open Car: Probably my second favorite rendition after the Arriving Somewhere… version. Nothing special–at least compared to “Lightbulb Sun”
  • Anesthetize: Really? I really wanted to love this. In fact, I love 99% of it. But at 7:13, he sings:

“The dust in my soul makes me feel the weight in my legs / my head in the clouds and I’m zoning out / my hand’s on a gun and I find the range, God, tempt me / I’m totally bored, but I can’t switch off”

The text in bold is supposed to be sung in the second verse. He should have sung “I’m watching TV, but I find it hard to stay conscious.” The rest of the song is fine. In fact, the rest of the song is great. But somehow this is one of those things that tickles even my subconscious because I know the song so well.

As an aside, I have similar feelings about “Futile” on Rockpalast. At about 2:15, right after the chorus, the correct line is: “You were the one collecting space.” What Wilson tries to sing is “You were the one just saving face,” which should be the second line. Instead, what comes out is “You were the one jollecting space.”

  • Blackest Eyes: Nothing special here.
  • Way Out Of Here: This one’s pretty solid; I didn’t really think much of this song until I’d heard it live, and this version does nothing to diminish my opinion of it. I have to say though, the Finns clapping at the beginning is fucking annoying. They clap a lot for every song, but this is ridiculous.
  • Sleep Together: Samesies, but the clapping is more acceptable.
  • Trains: Goddamn Finns fucking clapping at the fucking wrong time. I mean, really? There is exactly one segment of this song in which you are supposed to clap, and that segment is not the entire song. Fuck you, Finland.
  • Halo: Possibly my favorite “Halo” yet. I’m not a huge fan of the song, but, again, when I heard it live I thought it rocked. This one sounds a lot like the one I heard, so I dig it.

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Posted by Jay

Radioactive Toy = Weissensee?

April 5, 2009
neu_sunday_of_life

They even look alike!

Ok, maybe the title is a little misleading. Porcupine Tree and Neu! are very different bands. Obviously, Steven Wilson has found much inspiration in Neu!’s work. He notes in the album sleeve for Insignificance:

“Signify” was supposed to be P.T. play Krautrock, so much so that the demo actually starts off as a faithful (reasonably) cover version of the Neu! Classic “Hallogallo” (one of my top 5 desert island discs) – however clearly something went horribly wrong about 3½ minutes in and “Signify” was born

And the musical resemblance between “Hallogallo” and “Signify” is pretty obvious. But I was listening to Neu! the other day, and noticed a pretty strong resemblance between Neu!’s “Weissensee” and Porcupine Tree’s “Radioactive Toy.”

In particular, the base lines sound almost identical. They plod and pulse in basically the same way. Interestingly, I did a quick Google search, but no one else seems to be thinking about this.

Funny, that.

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Posted by Jay