analytic continuation

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1: 6.4 Analytic Continuation

§6.4 Analytic Continuation

►Analytic continuation of the principal value of $E_1\left(z\right)$ yields a multi-valued function with branch points at $z=0$ and $z=\mathrm{\infty }$. … ►

6.4.4 $$\mathrm{Ci}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \pi \mathrm{i}+\mathrm{Ci}\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E4

Encodings:

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See also:

Annotations for §6.4 and Ch.6

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6.4.7 $$\mathrm{g}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\mp \pi \mathrm{i}{\mathrm{e}}^{\mp \mathrm{i}z}+\mathrm{g}\left(z\right).$$

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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, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.12(ii), §6.4

Permalink:

http://dlmf.nist.gov/6.4.E7

Encodings:

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See also:

Annotations for §6.4 and Ch.6

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2: 28.7 Analytic Continuation of Eigenvalues

§28.7 Analytic Continuation of Eigenvalues

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28.7.4 $$\sum _{n=0}^{\mathrm{\infty }}\left(b_{2n+2}\left(q\right)-{(2n+2)}^2\right)=0.$$

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Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.7 and Ch.28

3: 14.24 Analytic Continuation

§14.24 Analytic Continuation

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4: 10.34 Analytic Continuation

§10.34 Analytic Continuation

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5: 10.11 Analytic Continuation

§10.11 Analytic Continuation

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6: 15.17 Mathematical Applications

… ►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. …

7: 16.15 Integral Representations and Integrals

… ►These representations can be used to derive analytic continuations of the Appell functions, including convergent series expansions for large $x$, large $y$, or both. …

8: 8.2 Definitions and Basic Properties

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§8.2(ii) Analytic Continuation

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9: 18.40 Methods of Computation

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Stieltjes Inversion via (approximate) Analytic Continuation

… ►The question is then: how is this possible given only $F_N(z)$, rather than $F(z)$ itself? $F_N(z)$ often converges to smooth results for $z$ off the real axis for $\mathrm{\Im }z$ at a distance greater than the pole spacing of the $x_n$, this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to $F_N(z)$ and evaluating these on the real axis in regions of higher pole density that those of the approximating function. …

10: 1.10 Functions of a Complex Variable

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§1.10(ii) Analytic Continuation

… ►If $f_2(z)$, analytic in $D_2$, equals $f_1(z)$ on an arc in $D=D_1\cap D_2$, or on just an infinite number of points with a limit point in $D$, then they are equal throughout $D$ and $f_2(z)$ is called an analytic continuation of $f_1(z)$. … ►Analytic continuation is a powerful aid in establishing transformations or functional equations for complex variables, because it enables the problem to be reduced to: (a) deriving the transformation (or functional equation) with real variables; followed by (b) finding the domain on which the transformed function is analytic. ►

Schwarz Reflection Principle

… ►Then the value of $F(z)$ at any other point is obtained by analytic continuation. …