# complex numbers

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## 1—10 of 135 matching pages

##### 1: 23.8 Trigonometric Series and Products
23.8.1 $\wp\left(z\right)+\frac{\eta_{1}}{\omega_{1}}-\frac{\pi^{2}}{4\omega_{1}^{2}}{% \csc}^{2}\left(\frac{\pi z}{2\omega_{1}}\right)=-\frac{2\pi^{2}}{\omega_{1}^{2% }}\sum_{n=1}^{\infty}\frac{nq^{2n}}{1-q^{2n}}\cos\left(\frac{n\pi z}{\omega_{1% }}\right),$
23.8.3 $\wp\left(z\right)=-\frac{\eta_{1}}{\omega_{1}}+\frac{\pi^{2}}{4\omega_{1}^{2}}% \sum_{n=-\infty}^{\infty}{\csc}^{2}\left(\frac{\pi(z+2n\omega_{3})}{2\omega_{1% }}\right),$
23.8.4 $\zeta\left(z\right)=\frac{\eta_{1}z}{\omega_{1}}+\frac{\pi}{2\omega_{1}}\sum_{% n=-\infty}^{\infty}\cot\left(\frac{\pi(z+2n\omega_{3})}{2\omega_{1}}\right),$
23.8.5 $\eta_{1}=\frac{\pi^{2}}{2\omega_{1}}\left(\frac{1}{6}+\sum_{n=1}^{\infty}{\csc% }^{2}\left(\frac{n\pi\omega_{3}}{\omega_{1}}\right)\right),$
##### 3: 23.2 Definitions and Periodic Properties
If $\omega_{1}$ and $\omega_{3}$ are nonzero real or complex numbers such that $\Im\left(\omega_{3}/\omega_{1}\right)>0$, then the set of points $2m\omega_{1}+2n\omega_{3}$, with $m,n\in\mathbb{Z}$, constitutes a lattice $\mathbb{L}$ with $2\omega_{1}$ and $2\omega_{3}$ lattice generators. …
23.2.11 $\zeta\left(z+2\omega_{j}\right)=\zeta\left(z\right)+2\eta_{j},$
where …
23.2.13 $\eta_{1}+\eta_{2}+\eta_{3}=0,$
23.2.15 $\sigma\left(z+2\omega_{j}\right)=-e^{2\eta_{j}(z+\omega_{j})}\sigma\left(z% \right),$
##### 4: 20.12 Mathematical Applications
The space of complex tori $\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ (that is, the set of complex numbers $z$ in which two of these numbers $z_{1}$ and $z_{2}$ are regarded as equivalent if there exist integers $m,n$ such that $z_{1}-z_{2}=m+\tau n$) is mapped into the projective space $P^{3}$ via the identification $z\to(\theta_{1}\left(2z\middle|\tau\right),\theta_{2}\left(2z\middle|\tau% \right),\theta_{3}\left(2z\middle|\tau\right),\theta_{4}\left(2z\middle|\tau% \right))$. …
##### 5: 1.1 Special Notation
In the physics, applied maths, and engineering literature a common alternative to $\overline{a}$ is $a^{*}$, $a$ being a complex number or a matrix; the Hermitian conjugate of $\mathbf{A}$ is usually being denoted $\mathbf{A}^{{\dagger}}$.
##### 6: 23.12 Asymptotic Approximations
Also,
23.12.4 $\eta_{1}=\frac{\pi^{2}}{4\omega_{1}}\left(\frac{1}{3}-8q^{2}+O\left(q^{4}% \right)\right),$
##### 7: 23.5 Special Lattices
The parallelogram $0$, $2\omega_{1}$, $2\omega_{1}+2\omega_{3}$, $2\omega_{3}$ is a square, and
23.5.2 $\eta_{1}=i\eta_{3}=\pi/(4\omega_{1}),$
23.5.6 $\eta_{1}=e^{\pi i/3}\eta_{3}=\frac{\pi}{2\sqrt{3}\omega_{1}},$
##### 8: 19.25 Relations to Other Functions
19.25.36 $\wp\left(z\right)-e_{j}\in\mathbb{C}\setminus(-\infty,0],$ $j=1,2,3.$
19.25.39 $\zeta\left(\omega_{j}\right)+\omega_{j}e_{j}=2R_{G}\left(0,e_{j}-e_{k},e_{j}-e% _{\ell}\right),$
##### 9: 23.10 Addition Theorems and Other Identities
23.10.12 $n\zeta\left(nz\right)=-n(n-1)(\eta_{1}+\eta_{3})+\sum_{j=0}^{n-1}\sum_{\ell=0}% ^{n-1}\zeta\left(z+\frac{2j}{n}\omega_{1}+\frac{2\ell}{n}\omega_{3}\right),$
23.10.13 $\sigma\left(nz\right)=A_{n}e^{-n(n-1)(\eta_{1}+\eta_{3})z}\prod_{j=0}^{n-1}% \prod_{\ell=0}^{n-1}\sigma\left(z+\frac{2j}{n}\omega_{1}+\frac{2\ell}{n}\omega% _{3}\right),$
23.10.15 $A_{n}=\left(\frac{\pi^{2}G^{2}}{\omega_{1}}\right)^{n^{2}-1}\frac{q^{n(n-1)/2}% }{i^{n-1}}\exp\left(-\frac{(n-1)\eta_{1}}{3\omega_{1}}\left((2n-1)(\omega_{1}^% {2}+\omega_{3}^{2})+3(n-1)\omega_{1}\omega_{3}\right)\right),$
##### 10: 31.2 Differential Equations
31.2.11 $\ifrac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}+\left(H+b_{0}\wp\left(\xi\right% )+b_{1}\wp\left(\xi+\omega_{1}\right)+b_{2}\wp\left(\xi+\omega_{2}\right)+b_{3% }\wp\left(\xi+\omega_{3}\right)\right)W=0,$