肯塔基社区暨科技学院成绩单《做证微fuk7778》iff

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1: 29.9 Stability

… ►If $\nu$ is not an integer, then (29.2.1) is unstable iff $h\le a_{\nu }^0\left(k^2\right)$ or $h$ lies in one of the closed intervals with endpoints $a_{\nu }^m\left(k^2\right)$ and $b_{\nu }^m\left(k^2\right)$, $m=1,2,\mathrm{\dots }$. If $\nu$ is a nonnegative integer, then (29.2.1) is unstable iff $h\le a_{\nu }^0\left(k^2\right)$ or $h\in [b_{\nu }^m\left(k^2\right),a_{\nu }^m\left(k^2\right)]$ for some $m=1,2,\mathrm{\dots },\nu$.

2: 1.7 Inequalities

… ►Equality holds iff $a_j=c{b}_j$, $\forall j$; $c=\text{ constant}$. … ►Equality holds iff $a_j=c{b}_j$, $\forall j$; $c=\text{ constant}$. … ►Equality holds iff $Af(x)=Bg(x)$ for all $x$. … ►Equality holds iff for all $x$. … ►Equality holds iff $Af(x)=Bg(x)$ for all $x$. …

3: 15.17 Mathematical Applications

… ►These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …

4: 32.10 Special Function Solutions

… ► ${\text{P}}_{\text{II}}$ has solutions expressible in terms of Airy functions (§9.2) iff … ► ${\text{P}}_{\text{III}}$ then has solutions expressible in terms of Bessel functions (§10.2) iff … ► ${\text{P}}_{\text{IV}}$ has solutions expressible in terms of parabolic cylinder functions (§12.2) iff either … ► ${\text{P}}_{\text{V}}$ then has solutions expressible in terms of Whittaker functions (§13.14(i)), iff … ► ${\text{P}}_{\text{VI}}$ has solutions expressible in terms of hypergeometric functions (§15.2(i)) iff …

5: 1.12 Continued Fractions

… ►A sequence $\{C_n\}$ in the extended complex plane, $ℂ\cup \{\mathrm{\infty }\}$, can be a sequence of convergents of the continued fraction (1.12.3) iff … ►The even part of $C$ exists iff $b_{2k}\ne 0$, $k=1,2,\mathrm{\dots }$, and up to equivalence is given by …The odd part of $C$ exists iff $b_{2k+1}\ne 0$, $k=0,1,2,\mathrm{\dots }$, and up to equivalence is given by … ►The continued fraction converges iff, in addition, …

6: 19.24 Inequalities

… ►with equality iff $a_0=g_0$. … ►All variables are positive, and equality occurs iff all variables are equal. … ►with equality iff $y=z$.

7: 28.29 Definitions and Basic Properties

… ►iff ${\mathrm{e}}^{\pi \mathrm{i}\nu }$ is an eigenvalue of the matrix … ►In consequence, (28.29.1) has a solution of period $\pi$ iff $\lambda ={\lambda }_n$, and a solution of period $2\pi$ iff $\lambda ={\mu }_n$. …

8: 32.8 Rational Solutions

… ►Then ${\text{P}}_{\text{III}}$ has rational solutions iff … ►In general, ${\text{P}}_{\text{IV}}$ has rational solutions iff either … ►Then ${\text{P}}_{\text{V}}$ has a rational solution iff one of the following holds with $m,n\in \mathbb{Z}$ and $\epsilon =\pm 1$: …

9: Guide to Searching the DLMF

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Table 3: A sample of recognized symbols

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Symbols	Comments
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->, <-, <->, =>, <==, <=>	For arrows $\to$, $\leftarrow$, $\leftrightarrow$, $\Rightarrow$, $\Leftarrow$ and $\iff$
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10: 23.20 Mathematical Applications

… ►The curve $C$ is made into an abelian group (Macdonald (1968, Chapter 5)) by defining the zero element $o=(0,1,0)$ as the point at infinity, the negative of $P=(x,y)$ by $-P=(x,-y)$, and generally $P_1+P_2+P_3=0$ on the curve iff the points $P_1$, $P_2$, $P_3$ are collinear. It follows from the addition formula (23.10.1) that the points $P_j=P(z_j)$, $j=1,2,3$, have zero sum iff $z_1+z_2+z_3\in 𝕃$, so that addition of points on the curve $C$ corresponds to addition of parameters $z_j$ on the torus $ℂ/𝕃$; see McKean and Moll (1999, §§2.11, 2.14). …