contour integrals on 1/z^k

some results.txt

On the line segment joining 1 to i,
the contour integral of 1/z^k for various powers of k works out to:

Integral(1/z^1) <= 2
Integral(1/z^2) <= 2*sqrt(2)
Integral(1/z^3) <= 4
Integral(1/z^4) <= 4*sqrt(2)
Integral(1/z^5) <= 8
Integral(1/z^6) <= 8*sqrt(2)
...
ie. Integral of (1/z^k) <= 2^(n+1)*sqrt(2), where k = 2*n+1

Proof: If you join 1+0i to 0+1i, the closest point to the origin is the midpoint ie. (1/2 + i/2)
Since contour integrals are bounded by ML, M = modulus of f(z), L = length of contour.
The length of contour ie. the line joining 1 to i, is simply sqrt(2)
M has max value at the midpoint (1/2+ i/2)
So regardless of k, L = sqrt(2)
M obviously depends on k
| 1/ (1/2 + i/2) | <= sqrt(2)
| 1/ (1/2 + i/2)^2 | <= 2
| 1/ (1/2 + i/2)^3 | <= 2*sqrt(2)
etc.
qed