Inner Products of Harmonic Functions

Skip to main content

\(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}} \renewcommand{\Hat}[1]{\mathbf{\boldsymbol{\hat{#1}}}} \let\VF=\vf \let\HAT=\Hat \newcommand{\Prime}{{}\kern0.5pt'} \newcommand{\PARTIAL}[2]{{\partial^2#1\over\partial#2^2}} \newcommand{\Partial}[2]{{\partial#1\over\partial#2}} \newcommand{\tr}{{\mathrm tr}} \newcommand{\CC}{{\mathbb C}} \newcommand{\HH}{{\mathbb H}} \newcommand{\KK}{{\mathbb K}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HR}{{}^*{\mathbb R}} \renewcommand{\AA}{\vf{A}} \newcommand{\BB}{\vf{B}} \newcommand{\CCv}{\vf{C}} \newcommand{\EE}{\vf{E}} \newcommand{\FF}{\vf{F}} \newcommand{\GG}{\vf{G}} \newcommand{\HHv}{\vf{H}} \newcommand{\II}{\vf{I}} \newcommand{\JJ}{\vf{J}} \newcommand{\KKv}{\vf{Kv}} \renewcommand{\SS}{\vf{S}} \renewcommand{\aa}{\VF{a}} \newcommand{\bb}{\VF{b}} \newcommand{\ee}{\VF{e}} \newcommand{\gv}{\VF{g}} \newcommand{\iv}{\vf{imath}} \newcommand{\rr}{\VF{r}} \newcommand{\rrp}{\rr\Prime} \newcommand{\uu}{\VF{u}} \newcommand{\vv}{\VF{v}} \newcommand{\ww}{\VF{w}} \newcommand{\grad}{\vf{\nabla}} \newcommand{\zero}{\vf{0}} \newcommand{\Ihat}{\Hat I} \newcommand{\Jhat}{\Hat J} \newcommand{\nn}{\Hat n} \newcommand{\NN}{\Hat N} \newcommand{\TT}{\Hat T} \newcommand{\ihat}{\Hat\imath} \newcommand{\jhat}{\Hat\jmath} \newcommand{\khat}{\Hat k} \newcommand{\nhat}{\Hat n} \newcommand{\rhat}{\HAT r} \newcommand{\shat}{\HAT s} \newcommand{\xhat}{\Hat x} \newcommand{\yhat}{\Hat y} \newcommand{\zhat}{\Hat z} \newcommand{\that}{\Hat\theta} \newcommand{\phat}{\Hat\phi} \newcommand{\LL}{\mathcal{L}} \newcommand{\DD}[1]{D_{\textrm{$#1$}}} \newcommand{\bra}[1]{\langle#1|} \newcommand{\ket}[1]{|#1\rangle} \newcommand{\braket}[2]{\langle#1|#2\rangle} \newcommand{\LargeMath}[1]{\hbox{\large$#1$}} \newcommand{\INT}{\LargeMath{\int}} \newcommand{\OINT}{\LargeMath{\oint}} \newcommand{\LINT}{\mathop{\INT}\limits_C} \newcommand{\Int}{\int\limits} \newcommand{\dint}{\mathchoice{\int\!\!\!\int}{\int\!\!\int}{}{}} \newcommand{\tint}{\int\!\!\!\int\!\!\!\int} \newcommand{\DInt}[1]{\int\!\!\!\!\int\limits_{#1~~}} \newcommand{\TInt}[1]{\int\!\!\!\int\limits_{#1}\!\!\!\int} \newcommand{\Bint}{\TInt{B}} \newcommand{\Dint}{\DInt{D}} \newcommand{\Eint}{\TInt{E}} \newcommand{\Lint}{\int\limits_C} \newcommand{\Oint}{\oint\limits_C} \newcommand{\Rint}{\DInt{R}} \newcommand{\Sint}{\int\limits_S} \newcommand{\Item}{\smallskip\item{$\bullet$}} \newcommand{\LeftB}{\vector(-1,-2){25}} \newcommand{\RightB}{\vector(1,-2){25}} \newcommand{\DownB}{\vector(0,-1){60}} \newcommand{\DLeft}{\vector(-1,-1){60}} \newcommand{\DRight}{\vector(1,-1){60}} \newcommand{\Left}{\vector(-1,-1){50}} \newcommand{\Down}{\vector(0,-1){50}} \newcommand{\Right}{\vector(1,-1){50}} \newcommand{\ILeft}{\vector(1,1){50}} \newcommand{\IRight}{\vector(-1,1){50}} \newcommand{\Partials}[3] {\displaystyle{\partial^2#1\over\partial#2\,\partial#3}} \newcommand{\Jacobian}[4]{\frac{\partial(#1,#2)}{\partial(#3,#4)}} \newcommand{\JACOBIAN}[6]{\frac{\partial(#1,#2,#3)}{\partial(#4,#5,#6)}} \newcommand{\LLv}{\vf{L}} \newcommand{\OOb}{\boldsymbol{O}} \newcommand{\PPv}{\vf{P}_\text{cm}} \newcommand{\RRv}{\vf{R}_\text{cm}} \newcommand{\ff}{\vf{f}} \newcommand{\pp}{\vf{p}} \newcommand{\tauv}{\vf{\tau}} \newcommand{\Lap}{\nabla^2} \newcommand{\Hop}{H_\text{op}} \newcommand{\Lop}{L_\text{op}} \newcommand{\Hhat}{\hat{H}} \newcommand{\Lhat}{\hat{L}} \newcommand{\defeq}{\overset{\rm def}{=}} \newcommand{\absm}{\vert m\vert} \newcommand{\ii}{\ihat} \newcommand{\jj}{\jhat} \newcommand{\kk}{\khat} \newcommand{\dS}{dS} \newcommand{\dA}{dA} \newcommand{\dV}{d\tau} \renewcommand{\ii}{\xhat} \renewcommand{\jj}{\yhat} \renewcommand{\kk}{\zhat} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

Section 14.5 Inner Products of Harmonic Functions

What happens if you multiply two different trig functions, for example \(\sin(2\theta)\sin(3\theta)\text{,}\) as shown in Figure 14.5. As you might expect, there are two wave structures superposed. The overall structure (often called an envelope) of a large wave controlled by the difference of the two wavelengths and with smaller “wiggles” controlled by the sum of the two wavelengths.

Figure 14.5. The graph of \(y=\sin(2\theta)\sin(3\theta)\text{.}\)

For Fourier series, what we will care about is the area under this graph. What is the integral of this combined function? Hard to tell, but there’s about as much area above the axis as below, so zero would be plausible guess, which turns out to be correct. This central identity underlies all of Fourier theory.

It is in fact not that difficult to evaluate such integrals in closed form. Using technology or integral tables, we find the indefinite integral

\begin{align} \int \cos(m\theta) \cos(n\theta) \,d\theta \amp = \frac{m\,\sin(m\theta)\cos(n\theta)-n\,\cos(m\theta)\sin(n\theta)} {m^2-n^2}\tag{14.5.1} \end{align}

for \(m\ne n\text{.}\) There are similar formulas for the various combinations of sines and cosines. You should notice that we will need to be careful when \(m\ne n\text{.}\)

For Fourier series, we care about the cases where \(m\) and \(n\) are both integers, and we want the definite integral over an entire period; with these assumptions, we find

\begin{align} \int_0^{2\pi} \sin(m\theta) \sin(n\theta) \,d\theta \amp = \delta_{mn} ,\tag{14.5.2}\\ \int_0^{2\pi} \sin(m\theta) \cos(n\theta) \,d\theta \amp = 0 ,\tag{14.5.3}\\ \int_0^{2\pi} \cos(m\theta) \cos(n\theta) \,d\theta \amp = \delta_{mn}\text{.}\tag{14.5.4} \end{align}

where the Kronecker delta \(\delta_{mn}\) is defined in Section 16.1.

To complete this list, we consider the case \(m=0\text{.}\) (Since \(\sin(0\,\theta)\) is the zero function, we do not need to include that case.) We get one last formula:

\begin{equation} \int_0^{2\pi} \cos^2(0\,\theta) \,d\theta = 2\pi\text{.}\tag{14.5.5} \end{equation}

Activity 14.3. Integrals of products of harmonic functions.

Use Euler’s formula (2.6.1) to find the values of the definite integrals in this section by hand, by explicitly integrating exponential functions.