Example of a complex transcendental number?

Question

Researching transcendental numbers I have only come across ones with a transcendental real part. I can't think of any which are pure imaginary or are not based on a real transendental number, t, of the form $t + ni$. Any ideas?

Edit: After some excellent responses, I now have another thought see Complex transcendentals not known in component form?

Answer 1

We have $a + bi$ is algebraic iff $a$ and $b$ are algebraic.

Therefore, if $a + bi$ is transcendental then at least one of $a$ or $b$ is transcendental.

So, all complex transcendental numbers are "based" on real transcendental numbers.

Answer 2

Real transcendental number: $\pi$, $e$, $\ldots$
Imaginary transcendental number: $i\pi$, $ie$, $ie^\pi$ , $i\pi^e (\text{suspected to be true but not yet proved})$ $\ldots$
Complex number with transcendental real and imaginary parts: $\pi+i\pi$, $e+i\pi$, $\pi+ie$, $e+ie$, $\ldots$

Moreover, a purely imaginary transcendental number can be $\ln(-1)$, following from the equation $e^{i\pi}=-1$ and in which case, we have to extend the domain of definition and hence the range of the natural logarithm.

Answer 3

I've always been intrigued by $$i^i=e^{-π/2}$$ (at least its principal value) and you will note that $$i^{i^i}=e^{(iπ/2) e^{-π/2}}$$ does not fall into the facile category of $a+ib$ where $a$ and $b$ are well-known transcendental numbers.