Bugs #9

Zoom out by a factor of 2 from yesterday’s image, we see the rope/web supports extend above and below the main body.

The rest of this post is for the more mathematically inclined readers. Yesterday I stated that the bug function is close to the identity. The real part is already there, so let’s focus on the imaginary part, and treat just the imaginary part as a real valued function.

Here is a table of values for three function and their first few derivatives evaluated at 0.

 f f(0) f'(0) f”(0) f”'(0) f””(0) f””'(0) x 0 1 0 0 0 0 sin(x) 0 1 0 -1 0 1 a*sin(x/a) 0 1 0 $-1/a^2$ 0 $1/a^4$

Notice that for the parameterized sin function, a*sin(x/a), with a large value of a, all of the derivatives are close to the derivatives of the identity function, x. UsingĀ  Taylor series, the difference can be made arbitrarily small of an arbitrary large neighborhood of the origin.

For example, $text{ for } epsilon, R text{ where }0 < e < 1 < R, text{ set }a = R^2/epsilon$ and use Taylor series to show that

$|x-a*sin(x/a)| < epsilon text{ whenever } |x| < R$

The same result carries over to the complex bug function.

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