Computing average values of complex numbers from a to b
I'm not sure whether we can talk about average values of imaginary numbers using integrals or not. I thought what would be the average value of $e^x$ from $-i$ to $i$. We know that average value of a function from $a$ to $b$ is:
$\frac{\int_a^bf(x)dx}{b - a}$
And if we let $f(x)$ be $e^x$ and $a = -i,b = i$
We might write the average value of the e^x from $-i$ to $ias following:
$avg=\frac{\int_{-i}^{+i}e^xdx}{i - (-i)}= \frac{1}{2i}[e^x]^{+i}_{-i}=\frac{1}{2i}(e^{i}-e^{-i})=\frac{e^i-e^{-i}}{2i} = sin(1)$
I thought it is pretty interesting to find result $sin(1)$ but the real question is:
1) Does this integral even make any sense? What happens if we try to integrate between two imaginary numbers ?
2) Is the average value of complex numbers defined? If so, is it still the same as real numbers ?
Thanks for your answers.