# Fungrim entry: e47bfb

Image: X-ray of $\theta_{2}\!\left(z , \frac{1}{2} i\right)$ on $z \in \left[-2, 2\right] + \left[-2, 2\right] i$
An X-ray plot illustrates the geometry of a complex analytic function $f(z)$. Thick black curves show where $\operatorname{Im}\!\left(f(z)\right) = 0$ (the function is pure real). Thick red curves show where $\operatorname{Re}\!\left(f(z)\right) = 0$ (the function is pure imaginary). Points where black and red curves intersect are zeros or poles. Magnitude level curves $\left|f(z)\right| = C$ are rendered as thin gray curves, with brighter shades corresponding to larger $C$. Blue lines show branch cuts. The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line. Yellow is used to highlight important regions.
Definitions:
Fungrim symbol Notation Short description
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
ConstI$i$ Imaginary unit
ClosedInterval$\left[a, b\right]$ Closed interval
Im$\operatorname{Im}(z)$ Imaginary part
Re$\operatorname{Re}(z)$ Real part
Abs$\left|z\right|$ Absolute value
Source code for this entry:
Entry(ID("e47bfb"),
Image(Description("X-ray of", JacobiTheta(2, z, Mul(Div(1, 2), ConstI)), "on", Element(z, Add(ClosedInterval(-2, 2), Mul(ClosedInterval(-2, 2), ConstI)))), ImageSource("xray_jacobi_theta_2_z_b")),
Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC