By thinking of the question in terms of vectors, any point lying on the locus, would satisfy that the vector from $i$ to $z$ would make an angle of $\pi/4$ with the horizontal line, like this:
But converting the equation into Cartesian co-ordinates, I get $y=x+1$, which as you know looks like so:
But if the part of the curve with negative x-values is included, then the $\theta$ would no longer be $\pi/4$, as shown here:
So by choosing a $z$ such that Re($z$)$<0$, arg($z$)$\neq \pi/4$
So my question is, which graph is correct, pink or green? And where have I gone wrong
Best Answer
In general, $$\arg(\alpha+i\beta)=\frac{\pi}{4}$$ implies $$\frac{\beta}{\alpha}=\arctan(\frac{\pi}{4})=1\Rightarrow \alpha=\beta,\color{red}{\text{ and }\alpha,\beta\ge0}$$ Assume $z=x+iy$. Therefore, $$z-i=x+iy-i=x+i(y-1)$$ Then $\arg(z-i)=\pi/4$ means the real part is equal to the imaginary part, i.e. $x=y-1$ Hence, $$y=x+1,\color{red}{\text{ and }x\ge0,y-1\ge0\Rightarrow y\ge1}$$ is the locus.