Can an indefinite integration of a real function with respect to real variable have a complex integral constant? i.e. for $\int f(x) dx = g(x) + c$, can '$c$' be complex, where $f(x)$ is a real function?
N.B. I have found a lot of integrations of real function (example 1, example 2) which use theorems from complex analysis, but finally get real value.
N.B. As per as I understand, it can't have complex constant, because integration of a real valued function would only be defined on real plane. Am I right?
N.B. I am just checking my understanding. I need to use this result in another problem.
Thanks in advance. 🙂
Best Answer
Sure, it can be complex: The indefinite integral $$\int f(x) dx=g(x)+C$$ is the antiderivative of the integrand $f(x)$. What this implies is that if you differentiate $g(x)+C$, you should get back to $f(x)$, and you will do this for any $C$, complex or not, as long as it isn't a function of $x$.