Integrate a complex exponential from negative infinity to infinity Why is the integral $$ \int_{-\infty}^{\infty} e^{i \theta} d\theta $$ not defined if each bound of integration can be written as the limit of an integer number $N$ of $\pm 2 \pi$ radians as the integer approaches infinity via
$$ \lim_{N\rightarrow\infty}(\pm 2 \pi N) = \pm \infty$$ ?
 A: By definition, $ \int_{-\infty}^\infty f(x)\; dx$ is the limit of $\int_{A}^B f(x)\; dx$ as $A \to -\infty$ and $B \to \infty$.  That is, if the value of the integral is $L$,
for every $\epsilon > 0$ there must be $M$ and $N$ such that whenever $A < -M$ and $B > N$, $$\left|L - \int_A^B f(x)\; dx\right| < \epsilon$$
In this case, $$\int_A^B \exp(i\theta)\; d\theta = i (e^{iA} - e^{iB})$$
For any $M$ and $N$, there are $A < -M$ and $B > N$ that make this $2 i$
and others that make this $-2i$, so there is no such $L$.
A: One could define
$$\displaystyle
\int_{-\infty}^\infty e^{i\theta}\;d\theta\tag{1}
$$
as
$$\displaystyle
\lim_{N\to\infty}\int_{-2N\pi}^{2N\pi}e^{i\theta}\;d\theta.
$$
But this is not very useful: one may have other interpretations of (1):
$$
\lim_{N\to\infty}\int_{-2N\pi}^{2(N+1)\pi}e^{i\theta}\;d\theta\,,\quad 
\lim_{N\to\infty}\int_{-2(N+1)\pi}^{2N\pi}e^{i\theta}\;d\theta\,,\quad
etc.
$$
They all have different values.
A major problem of (1) is that it is not absolutely  integrable:
$$
\int_{-\infty}^\infty |e^{i\theta}|\;d\theta=\infty
$$
In general, for function $f$ that is not absolutely integrable (over $\mathbb{R}$), the notion of $\int_{-\infty}^{\infty} f(x)\;dx$ is not very useful. Though, one may still study notions such as Cauchy principal value, or the one-sided improper Rieman integral.
