# Illustrating a contour path on the complex plane given parametrization

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.

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.