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: 1.5 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 $x\in I_1$. …

5: 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 …

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: 18.2 General Orthogonal Polynomials

… ►Here $w(x)$ is continuous or piecewise continuous or integrable such that … ►This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)). … ►The measure is not necessarily absolutely continuous (i. … ►Nevai (1979, p.39) defined the class $𝒮$ of orthogonality measures with support inside $[-1,1]$ such that the absolutely continuous part $w(x)dx$ has $w$ in the Szegő class $𝒢$. … ►

Monotonic Weight Functions

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9: 10.43 Integrals

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§10.43(i) Indefinite Integrals

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§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

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10: 18.18 Sums

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§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 $(-\mathrm{\infty },\mathrm{\infty })$. … ►

Expansion of $L^2$ functions

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