# square-integrable function

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## 5 matching pages

##### 1: 1.4 Calculus of One Variable
###### Square-IntegrableFunctions
A function $f(x)$ is square-integrable if …
##### 2: 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 …
##### 3: 30.4 Functions of the First Kind
If $f(x)$ is mean-square integrable on $[-1,1]$, then formally …
##### 4: 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).

• ##### 5: 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 $x\to\infty$. …