zeros of analytic functions

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1: 3.8 Nonlinear Equations

§3.8 Nonlinear Equations

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§3.8(v) Zeros of Analytic Functions

►Newton’s rule is the most frequently used iterative process for accurate computation of real or complex zeros of analytic functions $f(z)$. … ►

§3.8(vi) Conditioning of Zeros

… ►Corresponding numerical factors in this example for other zeros and other values of $j$ are obtained in Gautschi (1984, §4). …

2: 4.14 Definitions and Periodicity

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4.14.7 $$\cot z=\frac{\cos z}{\sin z}=\frac{1}{\tan z}.$$

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Defines:

$\cot z$: cotangent function

Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.6

Referenced by:

§4.14, §4.45(i), §4.45(ii)

Permalink:

http://dlmf.nist.gov/4.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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3: 5.2 Definitions

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5.2.1 $$\mathrm{\Gamma }\left(z\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{z-1}dt,$$ $\mathrm{\Re }z>0$.

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Defines:

$\mathrm{\Gamma }\left(z\right)$: gamma function

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

6.1.1

Referenced by:

§10.43(iii), (25.11.27), (25.11.28), (25.5.6), (25.5.7), §5.9(i), §5.9(ii), §8.21(ii), (9.12.17)

Permalink:

http://dlmf.nist.gov/5.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

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4: 1.10 Functions of a Complex Variable

… ►An analytic function $f(z)$ has a zero of order (or multiplicity) $m$ ($\ge 1$) at $z_0$ if the first nonzero coefficient in its Taylor series at $z_0$ is that of ${(z-z_0)}^m$. … … ►then the product ${\prod }_{n=1}^{\mathrm{\infty }}(1+a_n(z))$ converges uniformly to an analytic function $p(z)$ in $D$, and $p(z)=0$ only when at least one of the factors $1+a_n(z)$ is zero in $D$. …

5: 4.28 Definitions and Periodicity

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Periodicity and Zeros

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6: 10.21 Zeros

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§10.21(ii) Analytic Properties

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7: Bibliography C

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The analyticity of cross-product Bessel function zeros. Proc. Cambridge Philos. Soc. 62, pp. 215–226.

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External Links:

MathReview (R. G. Langebartel), ZentralBlatt

Referenced by:

§10.21(x).

Encodings:

BibTeX

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8: 4.2 Definitions

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§4.2(iii) The Exponential Function

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9: 23.2 Definitions and Periodic Properties

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§23.2(i) Lattices

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§23.2(ii) Weierstrass Elliptic Functions

… ►The function $\sigma \left(z\right)$ is entire and odd, with simple zeros at the lattice points. … …

10: 2.4 Contour Integrals

… ►Now assume that $c>0$ and we are given a function $Q(z)$ that is both analytic and has the expansion … ►Assume that $p(t)$ and $q(t)$ are analytic on an open domain $𝐓$ that contains $𝒫$, with the possible exceptions of $t=a$ and $t=b$. … ►

In a neighborhood of $a$

2.4.11

$$p(t)=p(a)+\sum _{s=0}^{\mathrm{\infty }}{p}_s{(t-a)}^{s+\mu },$$

$$q(t)=\sum _{s=0}^{\mathrm{\infty }}{q}_s{(t-a)}^{s+\lambda -1},$$

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Symbols:

$\lambda$: constant, $a$: left endpoint, $p(t)$: analytic function, $q(t)$: analytic function, $p_s$: coefficients, $q_s$: coefficients and $\mu$: positive constant

Referenced by:

§2.4(iv)

Permalink:

http://dlmf.nist.gov/2.4.E11

Encodings:

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See also:

Annotations for §2.4(iii), §2.4 and Ch.2

with $\mathrm{\Re }\lambda >0$, $\mu >0$, $p_0\ne 0$, and the branches of ${(t-a)}^{\lambda }$ and ${(t-a)}^{\mu }$ continuous and constructed with $\mathrm{ph}\left(t-a\right)\to \omega$ as $t\to a$ along $𝒫$.

… ►in which $z$ is a large real or complex parameter, $p(\alpha ,t)$ and $q(\alpha ,t)$ are analytic functions of $t$ and continuous in $t$ and a second parameter $\alpha$. … ►The function $f(\alpha ,w)$ is analytic at $w=w_1(\alpha )$ and $w=w_2(\alpha )$ when $\alpha \ne \widehat{\alpha }$, and at the confluence of these points when $\alpha =\widehat{\alpha }$. …