### Author Topic: Q4  (Read 2316 times)

#### Victor Ivrii

• Elder Member
• Posts: 2563
• Karma: 0
##### Q4
« on: October 28, 2016, 09:02:21 AM »
Decompose into full Fourier series on interval $[-\pi,\pi]$ and sketch the graph of the sum of such Fourier series:

f(x)=|x|.

#### Roro Sihui Yap

• Full Member
• Posts: 30
• Karma: 16
##### Re: Q4
« Reply #1 on: October 28, 2016, 09:33:02 AM »
Since $f(x) = |x|$ is an even function, $b_n = 0 \ \ \forall n$
$a_0 = \frac{1}{\pi}\int_{-\pi}^\pi |x|\,dx = \pi$
$a_n = \frac{1}{\pi}\int_{-\pi}^\pi |x|\cos(nx) \,dx = \int_0^\pi \frac{2x}{\pi}\cos(nx) \,dx$
Integrating by parts
$a_n =\frac{2x}{n\pi}\sin(nx) \big|_{0}^{\pi}- \int_0^\pi \frac{2}{n\pi} \sin(nx) \,dx = \frac{2}{n^2\pi}\cos(nx)\big|_{0}^{\pi}$
$a_n = \begin{cases}\frac{-4}{n^2\pi} && n \ is \ odd \\0 && n \ is \ even\end{cases}$

$f(x) = \frac{\pi}{2} +\sum_{m=0}^\infty \frac{-4}{(2m+1)^2\pi}\cos((2m +1)x)$

#### Victor Ivrii

Everybody understands that on the graph bottoms are at $2n\pi$ and picks at $(2n+1)\pi$, $n\in \mathbb{Z}$.
{"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": 1, "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, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9894110560417175, "perplexity": 24654.405303919724}, "config": {"markdown_headings": true, "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-2021-49/segments/1637964363689.56/warc/CC-MAIN-20211209061259-20211209091259-00360.warc.gz"}