How can I depict $~u\left(r\right)=\frac{1}{r}\exp\left(ir\right)~$?

Question

I want to depict the graph of the following formula .

$$ u \left( r \right) = \frac{1}{ r } \exp\left(i r\right) $$

Needless to say as we depict a locus of $~ \exp\left(i r \right) ~$ , the circle with radius $~ 1 ~$ can be shown .

As we call $~ \frac{1}{ r } ~$ as an amplitude , then this amplitude becomes smaller as $~ r ~$ increases .

Hence I thought the graph is like as the below one .

Is this graph is close to the correct one ?

I even want to depict it using some software .

Answer 1

Assuming $r$ is only a real number then we can expand the equation, as $e^{ix}=cos(x)+i$ $sin(x)$,

So we get $u(r)= \frac{cos(r)}{r}+i\frac{sin(r)}{r}$, so if we plot the real component we get

and if we plot the imaginary component. (the plots where made using desmos)

hopefully that helps.