piecewise continuous functions

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1: 6.16 Mathematical Applications

… ►It occurs with Fourier-series expansions of all piecewise continuous functions. … …

2: 1.4 Calculus of One Variable

… ►For an example, see Figure 1.4.1 ►

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Figure 1.4.1: Piecewise continuous function on $[a,b)$. Magnify

… ►If ${\varphi }^{\prime }(x)$ is continuous or piecewise continuous, then …

3: 3.7 Ordinary Differential Equations

… ►Let $(a,b)$ be a finite or infinite interval and $q(x)$ be a real-valued continuous (or piecewise continuous) function on the closure of $(a,b)$. …

4: 1.8 Fourier Series

… ►If $a_n$ and $b_n$ are the Fourier coefficients of a piecewise continuous function $f(x)$ on $[0,2\pi ]$, then …

5: 1.5 Calculus of Two or More Variables

… ►A function $f(x,y)$ is piecewise continuous on $I_1\times I_2$, where $I_1$ and $I_2$ are intervals, if it is piecewise continuous in $x$ for each $y\in I_2$ and piecewise continuous in $y$ for each $x\in I_1$. …

6: 2.3 Integrals of a Real Variable

… ►In addition to (2.3.7) assume that $f(t)$ and $q(t)$ are piecewise continuous (§1.4(ii)) on $(0,\mathrm{\infty })$, and …

7: 1.14 Integral Transforms

… ►If $f(t)$ is continuous and $f^{\prime }(t)$ is piecewise continuous on $[0,\mathrm{\infty })$, then … ►If $f(t)$ is piecewise continuous, then … ►Also assume that $f^{(n)}(t)$ is piecewise continuous on $[0,\mathrm{\infty })$. … ►If $f(t)$ and $g(t)$ are piecewise continuous, then … ►If $f(t)$ is piecewise continuous on $[0,\mathrm{\infty })$ and the integral (1.14.47) converges, then …

8: 10.43 Integrals

… ►

§10.43(i) Indefinite Integrals

… ►

§10.43(iii) Fractional Integrals

… ► … ►The Kontorovich–Lebedev transform of a function $g(x)$ is defined as … ►

$g(x)$ is piecewise continuous and of bounded variation on every compact interval in $(0,\mathrm{\infty })$, and each of the following integrals

…

9: 18.18 Sums

… ►

§18.18(i) Series Expansions of Arbitrary Functions

… ►Alternatively, assume $f(x)$ is real and continuous and $f^{\prime }(x)$ is piecewise continuous on $(-1,1)$. … ►Assume $f(x)$ is real and continuous and $f^{\prime }(x)$ is piecewise continuous on $(0,\mathrm{\infty })$. … ►Assume $f(x)$ is real and continuous and $f^{\prime }(x)$ is piecewise continuous on $(-\mathrm{\infty },\mathrm{\infty })$. … ►For the modified Bessel function $I_{\nu }\left(z\right)$ see §10.25(ii). …

10: 18.2 General Orthogonal Polynomials

… ►A system (or set) of polynomials $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, is said to be orthogonal on $(a,b)$ with respect to the weight function $w(x)$ ($\ge 0$) if ►

18.2.1 $${\int }_a^b{p}_n(x)p_m(x)w(x)dx=0,$$ $n\ne m$.

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Symbols:

$dx$: differential, $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ , $a$: interval endpoint, $b$: interval endpoint and $x$: real variable

A&S Ref:

22.1.1

Referenced by:

§18.2(i), §18.2(iii), §18.5(iii)

Permalink:

http://dlmf.nist.gov/18.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.2(i), §18.2(i), §18.2 and Ch.18

►Here $w(x)$ is continuous or piecewise continuous or integrable, and such that for all $n$. … ►More generally than (18.2.1)–(18.2.3), $w(x)dx$ may be replaced in (18.2.1) by a positive measure $d\alpha (x)$, where $\alpha (x)$ is a bounded nondecreasing function on the closure of $(a,b)$ with an infinite number of points of increase, and such that for all $n$. … ►This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)). …