# square-integrable function

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##### 1: 1.4 Calculus of One Variable
###### Square-IntegrableFunctions
A function $f(x)$ is square-integrable if …
##### 2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Let $X=[a,b]$ or $[a,b)$ or $(a,b]$ or $(a,b)$ be a (possibly infinite, or semi-infinite) interval in $\mathbb{R}$ . For a Lebesgue–Stieltjes measure $\,\mathrm{d}\alpha$ on $X$ let $L^{2}\left(X,\,\mathrm{d}\alpha\right)$ be the space of all Lebesgue–Stieltjes measurable complex-valued functions on $X$ which are square integrable with respect to $\,\mathrm{d}\alpha$ ,
1.18.13 $c_{n}=\left\langle f,\phi_{n}\right\rangle=\int_{a}^{b}f(x)\overline{\phi_{n}(% x)}\,\mathrm{d}x,$ $f\in L^{2}\left(X\right)$ ,
1.18.14 $\int_{a}^{b}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\sum_{n=0}^{\infty}{\left|c_{% n}\right|}^{2},$ $f\in L^{2}\left(X\right)$ ,
1.18.52 $\left\langle\left(z-T\right)^{-1}f,f\right\rangle=\int_{\boldsymbol{\sigma}}{% \left|\widehat{f}(\lambda)\right|}^{2}\frac{\,\mathrm{d}\lambda}{z-\lambda},$ $f\in L^{2}\left(X\right)$ , $z\notin\boldsymbol{\sigma}_{c}$ ,
1.18.65 $f(x)=\int_{\boldsymbol{\sigma}_{c}}\widehat{f}(\lambda)\phi_{\lambda}(x)\,% \mathrm{d}\lambda+\sum_{\boldsymbol{\sigma}_{p}}\widehat{f}(\lambda_{n})\phi_{% \lambda_{n}}(x),$ $f(x)\in C(X)\cap L^{2}\left(X\right)$ .
##### 3: 28.30 Expansions in Series of Eigenfunctions
Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi]$ can be expanded in a uniformly and absolutely convergent series …
##### 4: 30.4 Functions of the First Kind
If $f(x)$ is mean-square integrable on $[-1,1]$, then formally …
##### 5: 33.22 Particle Scattering and Atomic and Molecular Spectra
###### §33.22(iv) Klein–Gordon and Dirac Equations
• Eigenstates using complex-rotated coordinates $r\to re^{\mathrm{i}\theta}$, so that resonances have square-integrable eigenfunctions. See for example Halley et al. (1993).

• ##### 6: 1.8 Fourier Series
Let $f(x)$ be an absolutely integrable function of period $2\pi$, and continuous except at a finite number of points in any bounded interval. …
###### §1.8(iii) Integration and Differentiation
when $f(x)$ and $g(x)$ are square-integrable and $a_{n},b_{n}$ and $a^{\prime}_{n},b^{\prime}_{n}$ are their respective Fourier coefficients. … Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $|f^{\prime\prime}(x)|$ are integrable over $(-\infty,\infty)$. … Suppose also that $f(x)$ is integrable on $[0,\infty)$ and $f(x)\to 0$ as $x\to\infty$. …