## Approximation of a non-holomorphic function [closed]

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

Hi,

I am unfortunately very new to analysis and this may be a simple question-- if so, please forgive my ignorance.

The function $f(z) = z\bar{z}$ is not holomorphic (since it contains the $\bar{z}$ variable... this much I think I know...). This expression pops up in some of my work, and it is troublesome, because it prevents my functions from being analytic. If possible, I would love to replace this function with a holomorphic approximation of it, but I unfortunately do not know how to go about constructing such an approximation... Can someone provide me with a brief overview of what I might need to know?

Thanks!

-
Intuitively, the absolute value function is very non-holomorphic. A holomorphic function is locally conformal (at lest where it has nonzero derivative), i.e. preserves angles locally, while the absolute value function sends the complex plane to the real line so sends all angles to $0$ or $\pi$. – Alex Becker Sep 9 at 3:52
So I doub't you'll have much luck trying to approximate it with a holomorphic function. – Alex Becker Sep 9 at 3:53

1
This question is not really appropriate for this site - try math.stackexchange.com instead. However, to sharpen Alex Becker's remarks, note that integrating any function holomorphic on the disc round the circle of radius 1/2 gives zero, while integrating the function $f(z)=z\overline{z}$ round the same circle gives a non-zero constant. So any "holomorphic" approximation to $\overline{z}$ can't be a very good one. – Yemon Choi Sep 9 at 4:08

2
Not sure what you're doing, but trying to approximate $\bar{z}$ by a holomorphic function sounds like the wrong approach. You should reconsider what you're doing. – Deane Yang Sep 9 at 4:09
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 7, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9097951054573059, "perplexity_flag": "middle"}