characteristic exponents

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1: 28.34 Methods of Computation

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§28.34(i) Characteristic Exponents

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2: 28.36 Software

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§28.36(ii) Characteristic Exponents and Eigenvalues

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3: 28.29 Definitions and Basic Properties

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§28.29(ii) Floquet’s Theorem and the Characteristic Exponent

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28.29.9

2\cos\left(\pi\nu\right)=w_{\mbox{\tiny I}}(\pi,\lambda)+w_{\mbox{\tiny II}}^{% \prime}(\pi,\lambda).

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\nu$ : complex parameter, $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution, $w_{\mbox{\tiny II}}(z;a,q)$ : fundamental solution and $\lambda$ : parameter
Referenced by:: §28.29(ii), §28.29(ii), §28.29(ii)
Permalink:: http://dlmf.nist.gov/28.29.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §28.29(ii), §28.29 and Ch.28

… ►Given

\lambda

together with the condition (28.29.6), the solutions

\pm\nu

of (28.29.9) are the characteristic exponents of (28.29.1). …

4: 28.2 Definitions and Basic Properties

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§28.2(iii) Floquet’s Theorem and the Characteristic Exponents

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28.2.16

\cos\left(\pi\nu\right)=w_{\mbox{\tiny I}}(\pi;a,q)=w_{\mbox{\tiny I}}(\pi;a,-% q).

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $q=h^{2}$ : parameter, $\nu$ : complex parameter, $a$ : parameter and $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution
A&S Ref:: 20.3.10 (in slightly different form)
Referenced by:: §28.12(i), §28.12(i), §28.12(ii), §28.2(iii), §28.2(iv), §28.2(v), item (c), §28.34(i)
Permalink:: http://dlmf.nist.gov/28.2.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §28.2(iii), §28.2 and Ch.28

… ►Either

\widehat{\nu}

\nu

is called a characteristic exponent of (28.2.1). …

5: Bibliography B

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W. Bühring (1994) The double confluent Heun equation: Characteristic exponent and connection formulae. Methods Appl. Anal. 1 (3), pp. 348–370.

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External Links:: ISSN 1073-2772, FullText, MathReview (W. Balser), ZentralBlatt
Referenced by:: §31.12.
Encodings:: BibTeX

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6: 3.1 Arithmetics and Error Measures

… ►A nonzero normalized binary floating-point machine number

x

is represented as …where

s

is equal to

1

0

, each

b_{j}

j\geq 1

, is either

0

1

b_{1}

is the most significant bit,

p

(

\in\mathbb{N}

) is the number of significant bits

b_{j}

b_{p-1}

is the least significant bit,

E

is an integer called the exponent,

b_{0}.b_{1}b_{2}\dots b_{p-1}

is the significand, and

f=.b_{1}b_{2}\dots b_{p-1}

is the fractional part. … ►Let

E_{{\rm min}}\leq E\leq E_{{\rm max}}

with

E_{{\rm min}}<0

and

E_{{\rm max}}>0

. For given values of

E_{{\rm min}}

E_{{\rm max}}

, and

p

, the format width in bits

N

of a computer word is the total number of bits: the sign (one bit), the significant bits

b_{1},b_{2},\dots,b_{p-1}

(

p-1

bits), and the bits allocated to the exponent (the remaining

N-p

bits). The integers

p

E_{{\rm min}}

, and

E_{{\rm max}}

are characteristics of the machine. …

7: Bibliography M

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mpmath (free python library)

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Notes:: Mpmath is a pure-Python library for multiprecision floating-point arithmetic. It provides an extensive set of transcendental functions, unlimited exponent sizes, complex numbers, interval arithmetic, numerical integration and differentiation, root-finding, linear algebra, and much more.
External Links:: mpmath
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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H. P. Mulholland and S. Goldstein (1929) The characteristic numbers of the Mathieu equation with purely imaginary parameter. Phil. Mag. Series 7 8 (53), pp. 834–840.

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External Links:: Document, ZentralBlatt
Referenced by:: §28.7.
Encodings:: BibTeX

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H. J. W. Müller (1962) Asymptotic expansions of oblate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 211, pp. 33–47.

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External Links:: MathReview (J. Meixner), ZentralBlatt
Referenced by:: §30.9(ii), §30.9(ii).
Encodings:: BibTeX

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H. J. W. Müller (1963) Asymptotic expansions of prolate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 212, pp. 26–48.

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External Links:: MathReview (J. Meixner), ZentralBlatt
Referenced by:: §30.9(i), §30.9(i).
Encodings:: BibTeX

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H. J. W. Müller (1966a) Asymptotic expansions of ellipsoidal wave functions and their characteristic numbers. Math. Nachr. 31, pp. 89–101.

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External Links:: Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.11, §29.7(i), §29.7(ii).
Encodings:: BibTeX

…

8: 2.7 Differential Equations

… ►Let

\alpha_{1}

\alpha_{2}

denote the indices or exponents, that is, the roots of the indicial equation … ►where

\lambda_{1}

\lambda_{2}

are the roots of the characteristic equation … ►See §2.11(v) for other examples. … ►The transformed differential equation either has a regular singularity at

t=\infty

, or its characteristic equation has unequal roots. … ►This is characteristic of numerically satisfactory pairs. …