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Weierstrass sigma function
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(Definition)
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Definition 1 Let $\Lambda\subset\Complex$ be a lattice. Let $\Lambda^{\ast}$ denote $\Lambda-\{ 0 \}$
- The Weierstrass sigma function is defined as the product $$\sigma(z;\Lambda)=z\prod_{w\in\Lambda^{\ast}}\left(1-\frac{z}{w}\right)e^{z/w+\frac{1}{2}(z/w)^2}$$
- The Weierstrass zeta function is defined by the sum $$\zeta(z;\Lambda)=\frac{\sigma'(z;\Lambda)}{\sigma(z;\Lambda)}=\frac{1}{z}+\sum_{w\in\Lambda^{\ast}}\left( \frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^2}\right)$$ Note that the Weierstrass zeta function is basically the derivative of the logarithm of the sigma function. The zeta function can be rewritten as: $$\zeta(z;\Lambda)=\frac{1}{z}-\sum_{k=1}^{\infty}\mathcal{G}_{2k+2}(\Lambda)z^{2k+1}$$ where $\mathcal{G}_{2k+2}$ is the Eisenstein series of weight $2k+2$
- The Weierstrass eta function is defined to be $$\eta(w;\Lambda)=\zeta(z+w;\Lambda)-\zeta(z;\Lambda), \text{for any } z\in\Complex$$ (It can be proved that this is well defined, i.e. $\zeta(z+w;\Lambda)-\zeta(z;\Lambda)$ only depends on $w$ . The Weierstrass eta function must not be confused with the Dedekind eta function.
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"Weierstrass sigma function" is owned by alozano.
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See Also: elliptic function, modular discriminant
| Other names: |
sigma function, zeta function, eta function |
| Also defines: |
Weierstrass sigma function, Weierstrass zeta function, Weierstrass eta function |
| Keywords: |
Weierstrass, sigma, eta, zeta |
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Cross-references: Dedekind eta function, well defined, weight, Eisenstein series, logarithm, derivative, sum, product, lattice
There are 6 references to this entry.
This is version 1 of Weierstrass sigma function, born on 2003-08-25.
Object id is 4650, canonical name is WeierstrassSigmaFunction.
Accessed 11050 times total.
Classification:
| AMS MSC: | 33E05 (Special functions :: Other special functions :: Elliptic functions and integrals) |
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Pending Errata and Addenda
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