# Continuous deformation of the path

1. Nov 19, 2007

### John O' Meara

Experiment with a family of paths with common endpoints, say $$z(t) = t + \iota a sin(t) 0 \leq \ t \ \leq \pi$$, with real parameter a. Integrate non-analytic functions (Re(z), Re(z^2), etc.) and explore how the result depends on a. Take analytic functions of your choice compare and comment.
Relevant equations:
$$\int_C f(z(t))\frac{dz}{dt} dt.\\$$
f(z) =Re(z) = x but x(t)=t; a=1.
$$\frac{dz(t)}{dt}=1 +\iota \cos t \ \mbox{therefore} \ \int_C f(z) dz = \int_0 ^{\pi}t(1 + \iota \cos t)dt \\$$
$$\mbox{with a=2} \ \int_C f(z)dz = \int_0 ^{\pi} t(\+2\iota \cos t)dt \ \\ \mbox{ with a =3} \int_Cf(z)dz= \ \int_0 ^{\pi} t(1+3\iota \cos t)dt \\$$
We now take a second non-analytic function f(z)=Re(z^2) and a=1 $$f(z)=t^2-\sin^2 t \ \int_C f(z)dz \ = \ \int(t^2-\sin^2t)(1+\iota \cos t)dt \\$$
$$\mbox{ with a=2} \ \Re{z^2}=t^2-4\sin^2 t \ \mbox{therefore} \ \int_C \Re{z^2}dz \ = \ \int_0 ^{\pi}(t^2-4\sin^2t)(1+2\iota \cos t)dt \\$$ ..............(v)
Now taking some analytic functions with a=1; $$f(z) =z^2 = (t^2+\iota\sin t)^2 \ \ z(t)=t +\iota\sin t \ \ \frac{dz}{dt}=1 + \iota\cos t \ \\ \int_C z^2 dz = \int_0 ^{\pi} (t+\iota\sin t )^2(1+\iota \cos t)dt \\$$ ....................(vi)
$$\mbox{with a=2 } \ z(t)=t+2\iota\sin t \ \ \frac{dz(t)}{dt} \ = \ 1 + 2\iota\cos t \ \ f(z(t)) \ = \ (t \ + \ 2\iota\sin t)^2 \ \ \int_C z^2 dz \ = \ \int_0 ^{\pi} (t+2\iota\sin t)^2(1+2\iota\cos t)dt\\$$ .......................(vii)
When I carry out the integration for integrals (vi) and (vii) I don't get the same answer for them though they are analytic functions and should be independent of the path. I think the integrals may be wrongly formulated. Thanks.

2. Nov 22, 2007

### John O' Meara

The questions are:Am I correct to say that Re(z)=t, when a=1?
Am I right to say that Re(z^2) = t^2 - 4sin^2(t), when a=2, in equation (v)?
Am I correct in putting z^2 =(t+isin(t))^2 when a=1, in equation (vi)?
Am I correct in putting z^2 =(t+2isin(t))^2 when a=2 in equation (vii)?
Thanks for the help.

3. Nov 23, 2007

Yes
Yes
Yes
Yes