$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ ###[5.1.B. Discussion: pointwise convergence of Fourier integrals and series](id:sect-5.2.B) Recall [Theorem 4.4.2](./S4.4.html#thm-4.4.2): 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{eq-5.1B.1} \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( k ) \cos ( k x) + B( k )\sin ( k x)\Bigr)\,d k \label{eq-5.1B.2} \end{equation} where $A( k )$ and $B( k )$ 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{eq-5.1B.3}\\\\ \int\_{-\infty}^\infty C( k )e^{i k x}\,d k . \label{eq-5.1B.4} \end{gather} Why and what remedy do we have? If we consider definition of the partial sum of (\ref{eq-5.1B.1}) and then rewrite in the complex form and similar deal with (\ref{eq-5.1B.4}) 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{eq-5.1B.5}\\\\ \lim\_{N\to \infty} \int\_{-N}^N C( k )e^{i k x}\,d k \label{eq-5.1B.6}. \end{gather} Meanwhile convergence in (\ref{eq-5.1B.3}) and (\ref{eq-5.1B.4}) 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{eq-5.1B.7}\\\\ \lim\_{M,N\to \infty} \int\_{-M}^N C( k )e^{i k x}\,d k \label{eq-5.1B.8} \end{gather} where $M,N$ tend to $\infty$ independently. So the remedy is simple: understand convergence as in (\ref{eq-5.1B.5}), (\ref{eq-5.1B.6}) rather than as in (\ref{eq-5.1B.7}), (\ref{eq-5.1B.8}). For integrals such limit is called *principal value* of integral and is denoted by \begin{equation\*} \operatorname{pv}\int\_{-\infty}^\infty G( k )\,d k . \end{equation\*} BTW similarly is defined \begin{equation\*} \operatorname{pv}\int\_{a}^b G( k )\,d k := \lim\_{\varepsilon\to +0} \Bigl(\int\_a^{c-\varepsilon}G( k )\,d k + \int\_{c+\varepsilon}^bG( k )\,d k \Bigr) \end{equation\*} if there is a singularity at $c\in (a,b)$.Often instead of vp is used original (due to Cauchy) vp (valeur principale) and some other notations. 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. --------------- [$\Uparrow$](../contents.html)  [$\uparrow$](./S5.1.html)  [$\Rightarrow$](./S5.2.html)