# differentiable

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## 1—10 of 31 matching pages

##### 1: 1.4 Calculus of One Variable

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►When this limit exists $f$ is

*differentiable*at $x$. … ►When $n\ge 1$, $f$ is*continuously differentiable*on $I$. … … ►###### Mean Value Theorem

… ►A continuously differentiable function is convex iff the curve does not lie below its tangent at any point. …##### 2: 1.6 Vectors and Vector-Valued Functions

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►when $f$ is continuously differentiable.
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►The curve $C$ is

*piecewise differentiable*if $\mathbf{c}$ is piecewise differentiable. … … ►For $x$, $y$, and $z$ continuously differentiable, the vectors … ►For $f$ and $g$ twice-continuously differentiable functions …##### 3: 4.12 Generalized Logarithms and Exponentials

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►Both $\varphi (x)$ and $\psi (x)$ are continuously differentiable.
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►For ${C}^{\mathrm{\infty}}$ generalized logarithms, see Walker (1991).
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##### 4: 2.3 Integrals of a Real Variable

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►converges for all sufficiently large $x$, and $q(t)$ is infinitely differentiable in a neighborhood of the origin.
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►If, in addition, $q(t)$ is infinitely differentiable on $[0,\mathrm{\infty})$ and
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►assume $a$ and $b$ are finite, and $q(t)$ is infinitely differentiable on $[a,b]$.
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►Assume that $q(t)$ again has the expansion (2.3.7) and this expansion is infinitely differentiable, $q(t)$ is infinitely differentiable on $(0,\mathrm{\infty})$, and each of the integrals $\int {\mathrm{e}}^{\mathrm{i}xt}{q}^{(s)}(t)dt$, $s=0,1,2,\mathrm{\dots}$, converges at $t=\mathrm{\infty}$, uniformly for all sufficiently large $x$.
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►
(a)
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On $(a,b)$, $p(t)$ and $q(t)$ are infinitely differentiable and ${p}^{\prime}(t)>0$.

##### 5: 1.13 Differential Equations

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►Let $W(z)$ satisfy (1.13.14), $\zeta (z)$ be any thrice-differentiable function of $z$, and
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1.13.18
$$U(z)={({\zeta}^{\prime}(z))}^{1/2}W(z).$$

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1.13.19
$$\frac{{d}^{2}U}{{d\zeta}^{2}}=\left({\dot{z}}^{2}H(z)-\frac{1}{2}\{z,\zeta \}\right)U.$$

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1.13.22
$$\{z,\zeta \}=-{(dz/d\zeta )}^{2}\{\zeta ,z\}.$$

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►As the interval $[a,b]$ is mapped, one-to-one, onto $[0,c]$ by the above definition of $t$, the integrand being positive, the inverse of this same transformation allows $\widehat{q}(t)$ to be calculated from $p,q,\rho $ in (1.13.31), $p,\rho \in {C}^{2}(a,b)$ and $q\in C(a,b)$.
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##### 6: 18.32 OP’s with Respect to Freud Weights

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►where $Q(x)$ is real, even, nonnegative, and continuously differentiable, where $x{Q}^{\prime}(x)$ increases for $x>0$, and ${Q}^{\prime}(x)\to \mathrm{\infty}$ as $x\to \mathrm{\infty}$, see Freud (1969).
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##### 7: 1.5 Calculus of Two or More Variables

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►The function $f(x,y)$ is

*continuously differentiable*if $f$, $\partial f/\partial x$, and $\partial f/\partial y$ are continuous,*and twice-continuously differentiable*if also ${\partial}^{2}f/{\partial x}^{2}$, ${\partial}^{2}f/{\partial y}^{2}$, ${\partial}^{2}f/\partial x\partial y$, and ${\partial}^{2}f/\partial y\partial x$ are continuous. … ►
1.5.6
$$\frac{{\partial}^{2}f}{\partial x\partial y}=\frac{{\partial}^{2}f}{\partial y\partial x}.$$

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►If $F(x,y)$ is continuously differentiable, $F(a,b)=0$, and $\partial F/\partial y\ne 0$ at $(a,b)$, then in a *neighborhood*of $(a,b)$, that is, an open disk centered at $a,b$, the equation $F(x,y)=0$ defines a continuously differentiable function $y=g(x)$ such that $F(x,g(x))=0$, $b=g(a)$, and ${g}^{\prime}(x)=-{F}_{x}/{F}_{y}$. … ►If $f$ is $n+1$ times continuously differentiable, then … ►Sufficient conditions for validity are: (a) $f$ and $\partial f/\partial x$ are continuous on a rectangle $a\le x\le b$, $c\le y\le d$; (b) when $x\in [a,b]$ both $\alpha (x)$ and $\beta (x)$ are continuously differentiable and lie in $[c,d]$. …##### 8: 2.8 Differential Equations with a Parameter

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►The expansions (2.8.11) and (2.8.12) are both uniform and differentiable with respect to $\xi $.
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►Again, $u>0$ and $\psi (\xi )$ is ${C}^{\mathrm{\infty}}$ on $({\alpha}_{1},{\alpha}_{2})$.
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►The expansions (2.8.15) and (2.8.16) are both uniform and differentiable with respect to $\xi $.
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►The expansions (2.8.25) and (2.8.26) are both uniform and differentiable with respect to $\xi $.
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►The expansions (2.8.29) and (2.8.30) are both uniform and differentiable with respect to $\xi $.
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##### 9: 1.9 Calculus of a Complex Variable

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►

###### Differentiation

►A function $f(z)$ is*complex differentiable*at a point $z$ if the following limit exists: … ►Conversely, if at a given point $(x,y)$ the partial derivatives $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, and $\partial v/\partial y$ exist, are continuous, and satisfy (1.9.25), then $f(z)$ is differentiable at $z=x+\mathrm{i}y$. … ►A function $f(z)$ is said to be*analytic*(*holomorphic*) at $z={z}_{0}$ if it is complex differentiable in a neighborhood of ${z}_{0}$. … ►An*arc*$C$ is given by $z(t)=x(t)+\mathrm{i}y(t)$, $a\le t\le b$, where $x$ and $y$ are continuously differentiable. …##### 10: 1.8 Fourier Series

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►If a function $f(x)\in {C}^{2}[0,2\pi ]$ is periodic, with period $2\pi $, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to ${f}^{\prime}(x)$.
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►Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $\left|{f}^{\prime \prime}(x)\right|$ are integrable over $(-\mathrm{\infty},\mathrm{\infty})$.
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