Fourier transform of Sin(x + Sin(x))

Does anybody happen to know what the Fourier transform of $f(x) = \sin(x + \sin(x))$ is?

More generally, what is the Fourier transform of $g(x) = \sin(\phi_1 x + \mu \sin(\phi_2 x))$? ($\phi_1$, $\phi_2$ and $\mu$ are constants.)

I'm pretty sure I saw these written down somewhere, but Wolfram|Alpha insists that it cannot determine the result...

-

1 Answer

Since $\sin(x+\sin(x))$ is an odd function with period $2 \pi$, its Fourier series is $$\sin(x+\sin(x)) = \sum_{n=1}^\infty c_n \sin(n x)$$ where $$\begin{eqnarray} c_n &=& \frac{1}{\pi} \int_{-\pi}^\pi \sin(n x) \sin(x+\sin(x)) \mathrm{d} x = \frac{2}{\pi} \int_{0}^\pi \sin(n x) \sin(x+\sin(x)) \mathrm{d} x \\ &=& \frac{1}{\pi} \int_{0}^\pi \cos((n-1)x - \sin(x) ) \mathrm{d} x - \frac{1}{\pi} \int_{0}^\pi \cos((n+1)x + \sin(x) ) \mathrm{d} x \\ &=& J_{n-1}(1) - J_{n+1}(-1) = J_{n-1}(1) + (-1)^{n} J_{n+1}(1) \end{eqnarray}$$ where the last line was evaluated using Bessel integrals.

-

I thought I remembered Bessel-J being in there somewhere... Is there an easy expression for the more complicated function? – MathematicalOrchid May 3 '12 at 8:54
{"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": 2, "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": false, "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": 0.9229908585548401, "perplexity_flag": "middle"}