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11: 12.11 Zeros

… ►For example, let the $s$th real zeros of $U(a,x)$ and $U^{\prime }(a,x)$, counted in descending order away from the point $z=2\sqrt{-a}$, be denoted by $u_{a,s}$ and $u_{a,s}^{\prime }$, respectively. …

12: 1.10 Functions of a Complex Variable

… ►If $-n$ is the first negative integer (counting from $-\mathrm{\infty }$) with $a_{-n}\ne 0$, then $z_0$ is a pole of order (or multiplicity) $n$. … ►where $N$ and $P$ are respectively the numbers of zeros and poles, counting multiplicity, of $f$ within $C$, and ${\mathrm{\Delta }}_C(\mathrm{ph}f(z))$ is the change in any continuous branch of $\mathrm{ph}\left(f(z)\right)$ as $z$ passes once around $C$ in the positive sense. … ►each location again being counted with multiplicity equal to that of the corresponding zero or pole. …

13: 18.21 Hahn Class: Interrelations

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Figure 18.21.1: Askey scheme. …(This is with the convention that the real and imaginary parts of the parameters are counted separately in the case of the continuous Hahn polynomials.) Magnify

14: DLMF Project News

error generating summary

15: 1.2 Elementary Algebra

… ►Thus $det(𝐀)$ is the product of the $n$ (counted according to their multiplicities) eigenvalues of $𝐀$. …Thus $\mathrm{tr}(𝐀)$ is the sum of the (counted according to their multiplicities) eigenvalues of $𝐀$. …

16: Bibliography D

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Pólya’s Theory of Counting. In Applied Combinatorial Mathematics, E. F. Beckenbach (Ed.), pp. 144–184.

ⓘ

External Links:

ISBN 0898741726

Referenced by:

§26.20.

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BibTeX

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17: 1.11 Zeros of Polynomials

… ►A polynomial of degree $n$ with real or complex coefficients has exactly $n$ real or complex zeros counting multiplicity. …

18: 3.8 Nonlinear Equations

… ►has $n$ zeros in $ℂ$, counting each zero according to its multiplicity. …

19: 26.12 Plane Partitions

… ►Different configurations are counted as different plane partitions. …

20: 18.39 Applications in the Physical Sciences

… ►Thus the two missing quantum numbers corresponding to EOP’s of order $1$ and $2$ of the type III Hermite EOP’s are offset in the node counts by the fact that all excited state eigenfunctions also have two missing real zeros. …