Difference or relation b/w $ \exp(\theta i) $ and $\ e^ {\theta i } $

Question

I wanted to know Difference or relation b/w $ \exp(\theta i) $ and $\ e^ {\theta i} $.

So, I know $ exp(x) = \sum_{n=0}^\infty \frac{x^n}{n!}$ and If we plug in $ x = \theta i $ and using taylor series, we get $ exp(\theta i ) = \cos(\theta) + \sin(\theta)i $ and this is usually written as $\ e^ {\theta i} $.

but,

To find $ 2 ^ {\theta i} $ We would normal write it as $ e ^ {\log(2)\theta i} => \cos(\log(2)\theta) + \sin(\log(2)\theta)i$ but $e$ in this case is $e = exp(1)$ not $exp(x)$. So, $ log(x) $ which we took in first step in the case is bascially $ log_{exp(1)}x $ and $e^{\theta i}$ is in fact $exp(1) ^ {\theta i} = (\sum_{i=0}^n \frac{1}{i!}) ^ {\theta i} $

So, How can we use the same result that we drived for $ exp(\theta i )$ for $e^{\theta i}$?

and are both below expressions are same? If yes how?

$$e^{\theta i} = exp(1) ^ {\theta i} = \biggr(\sum_{i=0}^n \frac{1}{i!}\biggr)^ {\theta i} $$ same as

$$ exp(\theta i) = \sum_{n=0}^\infty \frac{(\theta i)^n}{n!}$$

or to simplify by taking $ \theta = 1 $ is $exp(1) ^ {i} => (\sum_{i=0}^n \frac{1}{i!}) ^ {i} $ same as $ exp(x) = \sum_{n=0}^\infty \frac{i^n}{n!}$

and I don't think we can use $\exp(x) = e^x$ for the imaginary part)

Answer 1

$\DeclareMathOperator{\Log}{Log}$In my experience, it's safest to come at your question slightly differently, in a way that minimizes taking complex powers.

As you say, $\exp$ is an entire function defined by a power series. This is our "foundation stone".

If $b$ is complex and $\log$ is a branch of logarithm whose domain contains $b$, the induced exponential function with base $b$ is defined by $$ b^{z} = \exp(z\log b). $$ If $e = \exp(1)$ is in the domain of $\log$, this definition is compatible with $e^{z} = \exp(z)$ for all complex $z$ if and only if $\log e = 1$. This holds for the principal branch, with imaginary part between $-\pi$ and $\pi$, and in fact $e^{z}$ is often used as a notational shorthand for $\exp(z)$.

If we write "$b^{z}$" without explanation, the default assumption is we're using the principal branch $\Log$.

Although $\exp(z + w) = \exp(z)\exp(w)$ and $b^{z+w} = b^{z} b^{w}$ for all complex $z$ and $w$, note carefully that other "familiar algebraic identities" such as $(ab)^{z} = a^{z}b^{z}$ and $a^{zw} = (a^{z})^{w}$ are not generally true for complex $a$, $b$, $z$ and $w$.

Unless the exponents are integers, the first step in working with complex powers is to write them in terms of the definition above.