$\renewcommand{\Re}{\operatorname{Re}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$
In the previous Lecture 16 we wrote Fourier series in the complex form
\begin{equation} f(x)= \sum_{n=-\infty}^\infty c_n e^{\frac{i\pi nx}{l}} \label{equ-17.1} \end{equation} with \begin{equation} c_n= \frac{1}{2l}\int_{-l}^l f(x)e^{-\frac{i\pi n x}{l}}\,dx \qquad n=\ldots,-2, -1, 0,1,2,\ldots \label{equ-17.2} \end{equation} and \begin{equation} 2l\sum_{n=-\infty}^\infty |c_n|^2=\int_{-l}^l|f(x)|^2\,dx. \label{equ-17.3} \end{equation}
From this form we formally without any justification deduct Fourier integral.
First we introduce \begin{equation} \omega_n := \frac{\pi n}{l}\qquad \text{and}\qquad \Delta \omega_n = \omega_{n}-\omega_{n-1}= \frac{\pi}{l} \label{equ-17.4} \end{equation} and rewrite (\ref{equ-17.1}) as \begin{equation} f(x)= \sum_{n=-\infty}^\infty C(\omega_n) e^{i\omega_n x}\Delta \omega_n \label{equ-17.5} \end{equation} with \begin{equation} C(\omega)= \frac{1}{2\pi}\int_{-l}^l f(x)e^{-i\omega x}\,dx \label{equ-17.6} \end{equation} where we used $C(\omega_n) := c_n /(\Delta\omega_n)$; (\ref{equ-17.3}) should be rewritten as \begin{equation} \int_{-l}^l|f(x)|^2\,dx= 2\pi\sum_{n=-\infty}^\infty |C(\omega_n)|^2\Delta \omega_n. \label{equ-17.7} \end{equation} Now we formally set $l\to +\infty$; then integrals from $-l$ to $l$ in the right-hand expression of (\ref{equ-17.6}) and the left-hand expression of (\ref{equ-17.7}) become integrals from $-\infty$ to $+\infty$.
Meanwhile, $\Delta\omega_n \to +0$ and Riemannian sums in the right-hand expressions of (\ref{equ-17.5}) and (\ref{equ-17.7}) become integrals: \begin{equation} f(x)= \int_{-\infty}^\infty C(\omega ) e^{i\omega x}\,d \omega \label{equ-17.8} \end{equation} with \begin{equation} C(\omega)= \frac{1}{2\pi}\int_{-\infty}^\infty f(x)e^{-i\omega x}\,dx; \label{equ-17.9} \end{equation} (\ref{equ-17.3}) becomes \begin{equation} \int_{-\infty}^\infty |f(x)|^2\,dx= 2\pi\int_{-\infty}^\infty |C(\omega)|^2\,d\omega. \label{equ-17.10} \end{equation}
Definition 1. (\ref{equ-17.9}) gives us a Fourier transform of $f(x)$, it usually is denoted by "hat": \begin{equation} \hat{f}(\omega)= \frac{1}{2\pi}\int_{-\infty}^\infty f(x)e^{-i\omega x}\,dx; \tag{FT}\label{FT} \end{equation} sometimes it is denoted by "tilde" ($\tilde{f}$), and seldom just by a corresponding capital letter $F(\omega)$.
Definition 2. (\ref{equ-17.8}) is a Fourier integral aka inverse Fourier transform: \begin{equation} f(x)= \int_{-\infty}^\infty \hat{f}(\omega) e^{i\omega x}\,d \omega \tag{FI}\label{FI} \end{equation} aka \begin{equation} \check{F}(x)= \int_{-\infty}^\infty F(\omega) e^{i\omega x}\,d \omega \label{IFT}\tag{IFT} \end{equation}
Remark 1. (\ref{equ-17.10}) is known as Plancherel theorem \begin{equation} \int_{-\infty}^\infty |f(x)|^2\,dx=2\pi\int_{-\infty}^\infty |\hat{f}(\omega )|^2\,d\omega. \tag{PT}\label{PT} \end{equation}
Applying the same arguments but as in the previous Lecture 16 we can rewrite (\ref{equ-17.8})--(\ref{equ-17.10}) as \begin{equation} f(x)= \int_0^\infty \bigl( A(\omega) \cos(\omega x) +B(\omega ) \sin (\omega x)\bigr) \,d \omega \label{equ-17.11} \end{equation} with \begin{align} & A(\omega)= \frac{1}{\pi}\int_{-\infty}^\infty f(x)\cos (\omega x) \,dx, \label{equ-17.12}\\ & B(\omega)= \frac{1}{\pi}\int_{-\infty}^\infty f(x)\sin (\omega x) \,dx, \label{equ-17.13} \end{align} and \begin{equation} \int_{-\infty}^\infty |f(x)|^2\,dx= \pi\int_0^\infty \bigl( |A(\omega )|^2+|B(\omega )|^2\bigr)\,d\omega. \label{equ-17.14} \end{equation}
$A(\omega)$ and $B(\omega)$ are $\cos$- and $\sin$- Fourier transforms and
Therefore
Recall Theorem 15.2
Let $f$ be a piecewise continuously differentiable function. Then the Fourier series \begin{equation} \frac{a_0}{2}+\sum_{n=1}^\infty \Bigl(a_n\cos \bigl(\frac{\pi n x}{l}\bigr) + a_n\cos \bigl(\frac{\pi n x}{l}\bigr)\Bigr) \label{equ-17.19} \end{equation} converges to
(b) $\frac{1}{2}\bigl(f(x+0)+f(x-0)\bigr)$ if $x$ is internal point and $f$ is discontinuous at $x$.
Exactly the same statement holds for Fourier Integral in the real form \begin{equation} \int_0^\infty \Bigl(A(\omega) \cos (\omega x) + B(\omega)\sin (\omega x)\Bigr)\,d\omega \label{equ-17.20} \end{equation} where $A(\omega)$ and $B(\omega)$ are $\cos$-and $\sin$-Fourier transforms.
None of them however holds for Fourier series or Fourier Integral in the complex form: \begin{gather} \sum_{n=-\infty}^\infty c_n e^{i\frac{\pi n x}{l}},\label{equ-17.21}\\ \int_{-\infty}^\infty C(\omega)e^{i\omega x}\,d\omega. \label{equ-17.22} \end{gather}
Why and what remedy do we have? If we consider definition of the partial sum of (\ref{equ-17.19}) and then rewrite in the complex form and similar deal with (\ref{equ-17.22}) we see that in fact we should look at \begin{gather} \lim_{N\to \infty} \sum_{n=-N}^N c_n e^{i\frac{\pi n x}{l}}, \label{equ-17.23}\\ \lim_{N\to \infty} \int_{-N}^N C(\omega)e^{i\omega x}\,d\omega \label{equ-17.24}. \end{gather} Meanwhile convergence in (\ref{equ-17.21}) and (\ref{equ-17.22}) means more than this: \begin{gather} \lim_{M,N\to \infty} \sum_{n=-M}^N c_n e^{i\frac{\pi n x}{l}}, \label{equ-17.25}\\ \lim_{M,N\to \infty} \int_{-M}^N C(\omega)e^{i\omega x}\,d\omega \label{equ-17.26} \end{gather} where $M,N$ tend to $\infty$ independently. So the remedy is simple: understand convergence as in (\ref{equ-17.23}), (\ref{equ-17.24}) rather than as in (\ref{equ-17.25}), (\ref{equ-17.26}).
For integrals such limit is called essential value of integral and is denoted by \begin{equation*} \operatorname{ess}-\int_{-\infty}^\infty G(\omega)\,d\omega \end{equation*} or \begin{equation*} \operatorname{vrai}-\int_{-\infty}^\infty G(\omega)\,d\omega \end{equation*} BTW similarly is defined \begin{equation*} \operatorname{ess}-\int_{a}^b G(\omega)\,d\omega:= \lim_{\varepsilon\to +0} \Bigl(\int_a^{c-\varepsilon}G(\omega)\,d\omega+ \int_{c+\varepsilon}^bG(\omega)\,d\omega\Bigr) \end{equation*} if there is a singularity at $c\in (a,b)$.
This is more general than the improper integrals studied in the end of Calculus I (which in turn generalize Riemann integrals). Those who took Complex Variables encountered such notion.