For a function $f(x)$, if I graph $f(2x)$, then it would look like $f(x)$ but compressed by a factor of $2$. If I graph $f(\frac{x}{2})$, then it would be stretched. I am very curious to know whether we can say something similar graphically about $f(ix)$, where $i = \sqrt{-1}$.
For instance for $\cosh(x),$ the graph looks like a parabola. However as soon as I scale it by a factor of $i$, the graph becomes a cosine curve(since $\cosh(ix) = \cos(x)$),completely looking different and having different properties from the original function. I am just curious to see whether there is a way to explain what happens when we scale a function by a factor of $i$, since it appears to be unpredictable. I would appreciate your help. Thanks.
