Check out the post here.

A brief explanation

The post mentions the sphere as a "one-point compactification" of the (complex) plane (by adding a point at infinity). The property of the sphere being compact somehow makes it a little closer to being "finite" and therefore easier to handle. But to understand more precisely what all this means you need to take a good course in Complex Analysis or Topology.

When studying complex analysis, I thought that the theorems are simpler, more beautiful, and closer to the finite case than analogous theorems in Real Analysis. I don't know whether that is due to the relationship with the sphere, but I suspect it is so.

Here is an example: You know that a polynomial p(x) with real coefficients (and a finite degree) can be written as a product of factors of the form (x-a) where a is a zero of the polynomial. (The root a is of-course a complex number). Turns out, under certain conditions, we can write a function (which can be viewed as an infinite series) as an (infinite) product of its zeros. For example, consider this formula:

Euler's Product:

(The formula above taken from Wikipedia's entry on the Wallis Product.)

The formula looks nicer if you replace x by (pi)x. Then the expression on the left has zeros at +1, +2, +3, ... and -1, -2, -3, .... And on the right you get factors of the form (1-x/n)(1+x/n) which is zero for x = +n and -n.

In fact, the way we write the product is something to do with making the product "converge" (or make sense).

This formula is definitely something I will write about one day. I think I need to pick up a complex analysis book again...its been too long...and have almost forgotten the beautiful stuff I used to see everyday.

## Friday, February 13, 2009

## Monday, February 9, 2009

### The Modulus Function

Every differentiable function

is also continuous.

"This does not mean,"

the Modulus Function

points out,

"that every continuous function

is differentiable too."

Math Poettary #5

is also continuous.

"This does not mean,"

the Modulus Function

points out,

"that every continuous function

is differentiable too."

Math Poettary #5

### Not differentiable

A continuous function

Going along nicely

without lifting pencil

from paper.

Ouch!

This sharp point

got me,

its not differentiable.

Math Poettary #4

Going along nicely

without lifting pencil

from paper.

Ouch!

This sharp point

got me,

its not differentiable.

Math Poettary #4

## Sunday, February 8, 2009

### Continuous function

A continuous function,

draw it without

picking up pencil

from paper.

At all points,

the left hand limit

the same as

the right hand limit,

the same as

the value of the function

at the point.

At all points

its value

what it should be.

Math Poettary #3

draw it without

picking up pencil

from paper.

At all points,

the left hand limit

the same as

the right hand limit,

the same as

the value of the function

at the point.

At all points

its value

what it should be.

Math Poettary #3

## Wednesday, January 28, 2009

### Isosceles Triangle

An Isosceles Triangle

looks in the mirror,

and finds itself un-reversed.

"My base angles are equal,"

it says.

Math Poettary #2

looks in the mirror,

and finds itself un-reversed.

"My base angles are equal,"

it says.

Math Poettary #2

## Saturday, January 10, 2009

### Ambigram Poettery

I started another blog with some "poettery" along with Punya Mishra's ambigrams. These are comments on ambigrams he has made over the years. You can find them here: Ambigram Poettery.

I call it poettery, because it is the product of my pottering around the poetry format. Really, its not poetry in the sense of the word-- can't really say I don't really understand the sense of the word.

Ambigrams, though, are something that are really interesting, and people interested in math are likely to like them.

### Binary Pascal's Triangle

Binary Pascal's Triangle

Originally uploaded by GauravBhatnagar

"Bit by bit,

I understand,

the triangle of Sierpinski"

says Pascal?

Math Poettary #1

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