# representation as Weierstrass elliptic functions

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## 4 matching pages

##### 1: 23.11 Integral Representations

###### §23.11 Integral Representations

…##### 2: 23.6 Relations to Other Functions

###### §23.6(iii) General Elliptic Functions

►For representations of general elliptic functions (§23.2(iii)) in terms of $\sigma \left(z\right)$ and $\mathrm{\wp}\left(z\right)$ see Lawden (1989, §§8.9, 8.10), and for expansions in terms of $\zeta \left(z\right)$ see Lawden (1989, §8.11). …##### 3: Bibliography E

##### 4: Errata

The Olver hypergeometric function $\mathbf{F}(a,b;c;z)$, previously omitted from the left-hand sides to make the formulas more concise, has been added. In Equations (15.6.1)–(15.6.5), (15.6.7)–(15.6.9), the constraint $$ has been added. In (15.6.6), the constraint $$ has been added. In Section 15.6 Integral Representations, the sentence immediately following (15.6.9), “These representations are valid when $$, except (15.6.6) which holds for $$.”, has been removed.

The Weierstrass lattice roots ${e}_{j},$ were linked inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots ${e}_{j}$, and lattice invariants ${g}_{2}$, ${g}_{3}$, now link to their respective definitions (see §§23.2(i), 23.3(i)).

*Reported by Felix Ospald.*

The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

These equations were rewritten with the modulus (second argument) of the Jacobian elliptic function defined explicitly in the preceding line of text.

Originally a minus sign was missing in the entries for $\mathrm{cd}u$ and $\mathrm{dc}u$ in the second column (headed $z+K+i{K}^{\prime}$). The correct entries are $-{k}^{-1}\mathrm{ns}z$ and $-k\mathrm{sn}z$. Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions $\mathrm{sn},\mathrm{cn},\mathrm{dn}$, whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

$u$ | ||||||
---|---|---|---|---|---|---|

$z+K$ | $z+K+\mathrm{i}{K}^{\prime}$ | $z+\mathrm{i}{K}^{\prime}$ | $z+2K$ | $z+2K+2\mathrm{i}{K}^{\prime}$ | $z+2\mathrm{i}{K}^{\prime}$ | |

$\mathrm{\vdots}$ | ||||||

$\mathrm{cd}u$ | $-\mathrm{sn}z$ | $-{k}^{-1}\mathrm{ns}z$ | ${k}^{-1}\mathrm{dc}z$ | $-\mathrm{cd}z$ | $-\mathrm{cd}z$ | $\mathrm{cd}z$ |

$\mathrm{\vdots}$ | ||||||

$\mathrm{dc}u$ | $-\mathrm{ns}z$ | $-k\mathrm{sn}z$ | $k\mathrm{cd}z$ | $-\mathrm{dc}z$ | $-\mathrm{dc}z$ | $\mathrm{dc}z$ |

$\mathrm{\vdots}$ |

*Reported 2014-02-28 by Svante Janson.*