Let $γ$ be sum of two line segments connecting $-2$ with $iy$ and $iy$ with $2$, where $y$ is fixed parameter. Write explicit parametrization for $γ$.

Question

QUESTION

Let $\gamma$ be the sum of two line segments connecting $-2$ with $iy$ and $iy$ with $2$, where $y$ is a fixed parameter.

i. Write an explicit parametrization for $\gamma$.

ii. For every $y$, evaluate the integrals $\int_\gamma z dz$ and $\int_\gamma \bar{z} dz$. Which of the integrals is independent of $y$?

iii. Use ii. to show that the conclusion of Cauchy's theorem does not hold for $f(z)=\bar z$

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DOUBT

I am confused with $y$ parameter and the one in $z=x+iy$. I know how to perform line integrals. But I am not sure how to compute parts i. and ii. I know the iii. part, as the contour is not closed, thus the answer won't be $\color{Tomato}{\text{zero}}$.

I will appreciate any help.

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MY TRY
I guess the line integral from $-2$ to $iy$ and then $iy$ to $2$ becomes $\color{Tomato}{\text{zero}}$, using Cauchy Theorem minus the integral on the line segment $-2$ to $2$? Am I right?

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Answer 1

Hints:

An easy and pretty useful (many times) parametrization of the straight line from point $\;(a,b)\,\,\,\text{ to point}\;\;(c,d)\;$ in the plane (real or complex) is

$$\gamma(t) = t(c,d)+(1-t)(a,b)=(\,(1-t)a+tc\,,\,\,(1-t)b+td\,)\;,\;\;\;t\in[0,1]$$

Now, for example: with the second line segment, from $\;(0,y)\;$ to $\;(2,0)\;$ we have:

$$\int_{\gamma_2}\overline z\,dz=\int_0^1\left[\left(1-t)\cdot2+2\cdot0\,,\,(1-t)y+t\cdot0\right)\bullet(2,\,-y)\right]\,dt=$$

$$=\int_0^1\left(4(1-t)-(1-t)y^2\right)\,dt=\ldots$$

As for part (iii): that's either a mistake in the question or, perhaps, they want you to join $\;-2\to2\;$ on the real line and close the two line segments and check there for each of $\;z,\,\overline z\;$