F(z) = [1-cosh(z)] / z^3. its pole, order & residue

1. May 31, 2009

bobmerhebi

1. Show that f(z) = [1 - cosh(z)] / z3 has a pole as its singular point. Determine its order m & find the residue B.

2.

lim |f(z)| tends to $$\infty$$ as z tends to singular point

bm + bm+1(z-z0)+...+ b1(z-z0)m-1 + $$\sum$$$$^{\infty}_{n = 0}$$an(z-z0)n= (z-z0) m f(z)

3. f(z) = [1 - cosh(z)] / z3 = 1/z3 - cos(iz)/z3

that's all i could do. I don't know how to do the limit of the cos part. does it approach infinity ? I think NOt right ?

plz help asap.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. May 31, 2009

eok20

Expand out cosh(z) = cos(iz) using its Taylor series.

3. May 31, 2009

bobmerhebi

Doing this i get:

f(z) = 1 - (1 + z2/2 + z4 / 4! + z6 / 6! + ....)

= - $$\sum^{\infty}_{n = 1}$$z2n/(2n)! tends to 0 as z tends to 0

but this is not the limit we have for a pole. but its for a removable singular point. isn't it ?

Similar Discussions: F(z) = [1-cosh(z)] / z^3. its pole, order & residue
{"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": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.764968752861023, "perplexity": 2763.095953906972}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886110792.29/warc/CC-MAIN-20170822143101-20170822163101-00533.warc.gz"}