I have tried to know more about behavior of function which has the similar form of error function where the power is not integer as shown with the below integral , The below integral is converge and has a closed form defined by incomplet Gamma function as shown here , The value of the integral over positive real line is equal to $\frac 1 e \Gamma(\frac 1 e)$ ,The problem that i have accrossed this integral produces complex numerical values converse to error function . Then My question here is :
Why the numerical values of $\displaystyle\int\limits_{-a}^{a} e^{-x^{e}}\ \text{d}x $ are not real ? or it has real values only if the power of $x$ is a positive integer ?.
Note: $a$ is a real number
Here is $\exp(-x^e)$. First the real part, then the imaginary part.
Since the imaginary part looks like that, we can see that the imaginary part of $\int_{-1}^{1} \exp(-x^e)\;dx$ is negative.