# real and imaginary parts

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##### 1: 19.3 Graphics Figure 19.3.7: K ⁡ ( k ) as a function of complex k 2 for - 2 ≤ ℜ ⁡ ( k 2 ) ≤ 2 , - 2 ≤ ℑ ⁡ ( k 2 ) ≤ 2 . … Magnify 3D Help Figure 19.3.8: E ⁡ ( k ) as a function of complex k 2 for - 2 ≤ ℜ ⁡ ( k 2 ) ≤ 2 , - 2 ≤ ℑ ⁡ ( k 2 ) ≤ 2 . … Magnify 3D Help Figure 19.3.9: ℜ ⁡ ( K ⁡ ( k ) ) as a function of complex k 2 for - 2 ≤ ℜ ⁡ ( k 2 ) ≤ 2 , - 2 ≤ ℑ ⁡ ( k 2 ) ≤ 2 . … Magnify 3D Help Figure 19.3.10: ℑ ⁡ ( K ⁡ ( k ) ) as a function of complex k 2 for - 2 ≤ ℜ ⁡ ( k 2 ) ≤ 2 , - 2 ≤ ℑ ⁡ ( k 2 ) ≤ 2 . … Magnify 3D Help Figure 19.3.12: ℑ ⁡ ( E ⁡ ( k ) ) as a function of complex k 2 for - 2 ≤ ℜ ⁡ ( k 2 ) ≤ 2 , - 2 ≤ ℑ ⁡ ( k 2 ) ≤ 2 . … Magnify 3D Help
##### 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
7.9.1 $\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z=\cfrac{z}{z^{2}+\cfrac{\frac{1}{2}}{1+% \cfrac{1}{z^{2}+\cfrac{\frac{3}{2}}{1+\cfrac{2}{z^{2}+\cdots}}}}},$ $\Re z>0$,
7.9.2 $\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z=\cfrac{2z}{2z^{2}+1-\cfrac{1\cdot 2}{2% z^{2}+5-\cfrac{3\cdot 4}{2z^{2}+9-\cdots}}},$ $\Re z>0$,
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
with $x_{1},x_{2},x_{3}$ real constants, has differential
19.32.2 $\mathrm{d}z=-\frac{1}{2}\left(\prod_{j=1}^{3}(p-x_{j})^{-1/2}\right)\mathrm{d}p,$ $\Im p>0$; $0<\operatorname{ph}\left(p-x_{j}\right)<\pi$, $j=1,2,3$.
$z(x_{3})=R_{F}\left(x_{3}-x_{1},x_{3}-x_{2},0\right)=-iR_{F}\left(0,x_{1}-x_{3% },x_{2}-x_{3}\right).$
As $p$ proceeds along the entire real axis with the upper half-plane on the right, $z$ describes the rectangle in the clockwise direction; hence $z(x_{3})$ is negative imaginary. …
##### 6: 23.1 Special Notation
 $\mathbb{L}$ lattice in $\mathbb{C}$. … complex variable, except in §§23.20(ii), 23.21(iii). …
##### 7: 27.14 Unrestricted Partitions
The corresponding unrestricted partition function is denoted by $p\left(n\right)$, and the summands are called parts; see §26.9(i). … The number of partitions of $n$ into at most $k$ parts is denoted by $p_{k}\left(n\right)$; again see §26.9(i). … where $\varepsilon=\exp\left(\pi\mathrm{i}(((a+d)/(12c))-s(d,c))\right)$ and $s(d,c)$ is given by (27.14.11). …
27.14.17 $\Delta\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{12}\Delta\left(\tau% \right),$
The 24th power of $\eta\left(\tau\right)$ in (27.14.12) with $e^{2\pi\mathrm{i}\tau}=x$ is an infinite product that generates a power series in $x$ with integer coefficients called Ramanujan’s tau function $\tau\left(n\right)$: …
##### 8: 23.11 Integral Representations
provided that $-1<\Re\left(z+\tau\right)<1$ and $\left|\Im z\right|<\Im\tau$.
##### 9: 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 $\Im\psi\left(\tfrac{1}{2}+iy\right)=\frac{\pi}{2}\tanh\left(\pi y\right),$
##### 10: 4.2 Definitions
The real and imaginary parts of $\ln z$ are given by …