# imaginary part

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## 1—10 of 188 matching pages

##### 2: 29.16 Asymptotic Expansions
The approximations for Lamé polynomials hold uniformly on the rectangle $0\leq\Re z\leq K$, $0\leq\Im z\leq{K^{\prime}}$, when $nk$ and $nk^{\prime}$ assume large real values. …
##### 3: 7.9 Continued Fractions
7.9.3 $w\left(z\right)=\frac{i}{\sqrt{\pi}}\cfrac{1}{z-\cfrac{\frac{1}{2}}{z-\cfrac{1% }{z-\cfrac{\frac{3}{2}}{z-\cfrac{2}{z-\cdots}}}}},$ $\Im z>0$.
##### 4: 28.25 Asymptotic Expansions for Large $\Re z$
28.25.4 $\Re z\to+\infty,$ $-\pi+\delta\leq\operatorname{ph}h+\Im z\leq 2\pi-\delta$,
28.25.5 $\Re z\to+\infty,$ $-2\pi+\delta\leq\operatorname{ph}h+\Im z\leq\pi-\delta$,
##### 5: 19.32 Conformal Map onto a Rectangle
19.32.2 $\,\mathrm{d}z=-\frac{1}{2}\left(\prod_{j=1}^{3}(p-x_{j})^{-1/2}\right)\,% \mathrm{d}p,$ $\Im p>0$; $0<\operatorname{ph}\left(p-x_{j}\right)<\pi$, $j=1,2,3$.
##### 6: 23.1 Special Notation
 $\mathbb{L}$ lattice in $\mathbb{C}$. … complex variable, except in §§23.20(ii), 23.21(iii). … lattice generators ($\Im\left(\omega_{3}/\omega_{1}\right)>0$). … lattice parameter ($\Im\tau>0$). …
Whittaker and Watson (1927) requires only $\Im\left(\omega_{3}/\omega_{1}\right)\neq 0$, instead of $\Im\left(\omega_{3}/\omega_{1}\right)>0$. …
##### 7: 27.14 Unrestricted Partitions
27.14.12 $\eta\left(\tau\right)=e^{\pi\mathrm{i}\tau/12}\prod_{n=1}^{\infty}(1-e^{2\pi% \mathrm{i}n\tau}),$ $\Im\tau>0$.
27.14.13 $\eta\left(\tau\right)=e^{\pi\mathrm{i}\tau/12}\mathit{f}\left(e^{2\pi\mathrm{i% }\tau}\right).$
27.14.14 $\eta\left(\frac{a\tau+b}{c\tau+d}\right)=\varepsilon(-\mathrm{i}(c\tau+d))^{% \frac{1}{2}}\eta\left(\tau\right),$
27.14.16 $\Delta\left(\tau\right)=(2\pi)^{12}(\eta\left(\tau\right))^{24},$ $\Im\tau>0$,
27.14.17 $\Delta\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{12}\Delta\left(\tau% \right),$
##### 8: 20.14 Methods of Computation
For values of $\left|q\right|$ near $1$ the transformations of §20.7(viii) can be used to replace $\tau$ with a value that has a larger imaginary part and hence a smaller value of $\left|q\right|$. …In theory, starting from any value of $\tau$, a finite number of applications of the transformations $\tau\to\tau+1$ and $\tau\to-1/\tau$ will result in a value of $\tau$ with $\Im\tau\geq\sqrt{3}/2$; see §23.18. In practice a value with, say, $\Im\tau\geq 1/2$, $\left|q\right|\leq 0.2$, is found quickly and is satisfactory for numerical evaluation.
##### 9: 23.11 Integral Representations
provided that $-1<\Re\left(z+\tau\right)<1$ and $\left|\Im z\right|<\Im\tau$.
##### 10: 5.4 Special Values and Extrema
5.4.16 $\Im\psi\left(iy\right)=\frac{1}{2y}+\frac{\pi}{2}\coth\left(\pi y\right),$
5.4.17 $\Im\psi\left(\tfrac{1}{2}+iy\right)=\frac{\pi}{2}\tanh\left(\pi y\right),$