# Jordan curve theorem

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##### 1: Peter L. Walker

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►Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan.
►Walker’s books are

*An Introduction to Complex Analysis*, published by Hilger in 1974,*The Theory of Fourier Series and Integrals*, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and*Examples and Theorems in Analysis*, published by Springer in 2004. …##### 2: 27.2 Functions

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###### §27.2(i) Definitions

… ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ►If $\left(a,n\right)=1$, then the*Euler–Fermat theorem*states that …This is*Jordan’s function*. Note that ${J}_{1}\left(n\right)=\varphi \left(n\right)$. …##### 3: 22.18 Mathematical Applications

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###### §22.18(i) Lengths and Parametrization of Plane Curves

… ►###### §22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

►Algebraic curves of the form ${y}^{2}=P(x)$, where $P$ is a nonsingular polynomial of degree 3 or 4 (see McKean and Moll (1999, §1.10)), are*elliptic curves*, which are also considered in §23.20(ii). …The theory of elliptic functions brings together complex analysis, algebraic curves, number theory, and geometry: Lang (1987), Siegel (1988), and Serre (1973).##### 4: 1.9 Calculus of a Complex Variable

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###### DeMoivre’s Theorem

… ►###### Jordan Curve Theorem

… ►###### Cauchy’s Theorem

… ►###### Liouville’s Theorem

… ►###### Dominated Convergence Theorem

…##### 5: Bibliography J

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Tables of Functions with Formulae and Curves.
4th edition, Dover Publications, New York.
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Calculus of Finite Differences.
Hungarian Agent Eggenberger Book-Shop, Budapest.
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Calculus of Finite Differences.
3rd edition, AMS Chelsea, Providence, RI.
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##### 6: 27.6 Divisor Sums

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27.6.7
$$\sum _{d|n}\mu \left(d\right){\left(\frac{n}{d}\right)}^{k}={J}_{k}\left(n\right),$$

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27.6.8
$$\sum _{d|n}{J}_{k}\left(d\right)={n}^{k}.$$

##### 7: 23.20 Mathematical Applications

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###### §23.20(ii) Elliptic Curves

►An algebraic curve that can be put either into the form …is an example of an*elliptic curve*(§22.18(iv)). … ► $K$ always has the form $T\times {\mathbb{Z}}^{r}$ (*Mordell’s Theorem*: Silverman and Tate (1992, Chapter 3, §5)); the determination of $r$, the rank of $K$, raises questions of great difficulty, many of which are still open. … ► …##### 8: 28.27 Addition Theorems

###### §28.27 Addition Theorems

►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …##### 9: 24.17 Mathematical Applications

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►See Milne-Thomson (1933), Nörlund (1924), or Jordan (1965).
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###### §24.17(iii) Number Theory

►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)). …##### 10: 5.16 Sums

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