maximum-modulus principle

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1: 1.10 Functions of a Complex Variable

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Phase (or Argument) Principle

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§1.10(v) Maximum-Modulus Principle

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Analytic Functions

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Harmonic Functions

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Schwarz’s Lemma

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2: 3.8 Nonlinear Equations

… ►Initial approximations to the zeros can often be found from asymptotic or other approximations to $f(z)$, or by application of the phase principle or Rouché’s theorem; see §1.10(iv). …

3: 10.74 Methods of Computation

… ►To ensure that no zeros are overlooked, standard tools are the phase principle and Rouché’s theorem; see §1.10(iv). …

4: 27.10 Periodic Number-Theoretic Functions

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27.10.2 $$f(n)=\sum _{m=1}^{k}g(m){\mathrm{e}}^{2\pi \mathrm{i}mn/k},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer, $f(n)$: function and $g(m)$: periodic function

Permalink:

http://dlmf.nist.gov/27.10.E2

Encodings:

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See also:

Annotations for §27.10 and Ch.27

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27.10.3 $$g(m)=\frac{1}{k}\sum _{n=1}^{k}f(n){\mathrm{e}}^{-2\pi \mathrm{i}mn/k}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer, $f(n)$: function and $g(m)$: periodic function

Permalink:

http://dlmf.nist.gov/27.10.E3

Encodings:

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See also:

Annotations for §27.10 and Ch.27

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27.10.4 $$c_k\left(n\right)=\sum _{m=1}^k{\chi }_1\left(m\right){\mathrm{e}}^{2\pi \mathrm{i}mn/k},$$

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Defines:

$c_k\left(n\right)$: Ramanujan’s sum

Symbols:

$\chi \left(n\right)$: Dirichlet character, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E4

Encodings:

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See also:

Annotations for §27.10 and Ch.27

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27.10.7 $$s_k(n)=\sum _{m=1}^k{a}_k(m){\mathrm{e}}^{2\pi \mathrm{i}mn/k},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer and $a_k(m)$: coefficients

Permalink:

http://dlmf.nist.gov/27.10.E7

Encodings:

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Annotations for §27.10 and Ch.27

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27.10.9 $$G(n,\chi )=\sum _{m=1}^k\chi \left(m\right){\mathrm{e}}^{2\pi \mathrm{i}mn/k}.$$

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Defines:

$G(n,\chi )$: Gauss sum

Symbols:

$\chi \left(n\right)$: Dirichlet character, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E9

Encodings:

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See also:

Annotations for §27.10 and Ch.27

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5: Bibliography R

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Principles of Mathematical Analysis. 3rd edition, McGraw-Hill Book Co., New York.

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Notes:

International Series in Pure and Applied Mathematics

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.4(iii).

Encodings:

BibTeX

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6: Bibliography F

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Principles and Techniques of Applied Mathematics. Dover, New York.

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Notes:

Reprint of First Edition, 1956, John-Wiley: New York

External Links:

ISBN 0-486-66444-9, MathReview Entry

Referenced by:

§1.16(ix), §1.17(ii), §1.17(ii), §1.17(i), §1.17(i), §1.17(iv), §1.18(ix), §1.18(iii), §1.18(vi), §1.18(i), §1.18(x).

Encodings:

BibTeX

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7: 10.11 Analytic Continuation

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8: 18.39 Applications in the Physical Sciences

… ►which is the quantum superposition principle. … … ►where $n$ is now the Bohr Principle Quantum Number. …

9: 26.6 Other Lattice Path Numbers

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10: Bibliography L

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The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity. Methuen and Co., Ltd., London.

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Notes:

Translated from Das Relativitätsprinzip by W. Perrett and G. B. Jeffery. Reprinted by Dover Publications Inc., New York, 1952.

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.6(ii).

Encodings:

BibTeX

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