I need to find an upper bound of the following integral:
$$\left| \int_{\Gamma} \ln (z+3) \right|$$
Where $\Gamma$ is the line segment from
$(-1+3i)$ to $(4+3i)$.
We have:
$$\left| \int_{\Gamma} \ln (z+3) \right|~\leq ML$$
Finding L is easy, as it is just the distance between the given points, which in this this case is equal to:
$5$.
So:
$$\left| \int_{\Gamma} \ln (z+3) \right|~\leq 5M$$ Now, I'm a little lost when it comes to this:
$$ln(z+3)$$
I tried doing this:
$$ln(x+iy+3)$$
But I'm kinda lost as is not close to what the textbook I'm using has.
Any ideas?
Note that
$$ \ln(z+3) = |z+3| + i \arg(z+3) $$
with $z = x + 3i$ where $-1 \le x \le 4$ or $2 \le x+3 \le 7$
Then
$$ |z+3|^2 = (x+3)^2 + 3^2 \le 7^2 + 3^2 = 58 $$
$$ \arg(z+3) = \arctan \frac{3}{x+3} \le \arctan\frac32 $$
Finally, you can make a bound
$$ |ln(z+3)| = \sqrt{|z+3|^2 + \arg^2(z+3)} \le \sqrt{58 + \arctan^2 \frac32} $$