Illustrating a contour path on the complex plane given parametrization

Question

I have the following contour parametrization: $$f(t) = (5+3i) + (1+i)e ^{it} , t \in [0, \pi]$$

I need to illustrate the contour on the complex plane.

What I've done is calculate the two ends points of the contour, where $$ t=0$$ and $$t = \pi$$

I've got the points: $$ (6+4i), (4+2i)$$ accordingly.

Then I put these points on the complex plane and drew a line between the first and the second, where the direction is towards the second. Is this the correct answer? I've tried using Wolfram to check myself but I can't figure out how to view this specific contour. If this is wrong I'd appreciate an explanation on how to approach this problem.

Answer 1

The set of points of the form $e^{it}$, with $t\in[0,\pi]$, consists of the upper half of the unit circle. Multiplying this by $1+i$ is the same thing as multiplying it by$$\frac1{\sqrt2}+\frac i{\sqrt 2}\left(=\cos\left(\frac\pi4\right)+\sin\left(\frac\pi4\right)i\right)$$and then to multiply this by $\sqrt 2$. The first operation is a rotation around the origin in the direct sense, with an angle equal to $\frac\pi4$; the second operation is a homothety centered at the origin with augmentation factor $\sqrt2$. After having done this, you get an half-circle centered at the origin that goes from $1+i$ to $-1-i$. When you add $5+3i$ to this, you get a similar half-circle, centered at $5+3i$; see the picture below.