# Fourier transformation - connection between exponential and trigonomeric forms

On Wikipedia i have come across a Fourier transformation equation in exponential form and its inverse (Wiki):

$$\begin{split} \mathcal{F}(x) &= \int\limits^{\infty}_{-\infty}\mathcal{f}(k) \, e^{2 \pi i kx} \, \textrm{d} k\\ \mathcal{f}(k) &= \int\limits^{\infty}_{-\infty}\mathcal{F}(x) \, e^{2 \pi i kx} \, \textrm{d} x \end{split}$$

but i allso found that there is a trigonometric form of Fourier transformation (PDF, page 2)

$$\begin{split} \mathcal{F}(x) &= \int\limits^{\infty}_{-\infty} f(k) \cos(kx) \, \textrm{d}k\\ \mathcal{f}(k) &= \int\limits^{\infty}_{-\infty} \mathcal{F}(x) \cos(kx) \, \textrm{d}x \end{split}$$

MAIN QUESTION:

Could someone show me, how these pairs of equations are connected?

SUB QUESTION:

(i) I think that $\textrm{d}x$ is used for spatial integration (please correct me if i am wrong).

(ii) I think that $\textrm{d}k$ is used for integration over wave vector (please correct me if i am wrong) .

-

## 1 Answer

Do you know about Euler's Formula?

$e^{ix} = cos(x) + isin(x)$

I think that's a hint...

Also, see this formula http://en.wikipedia.org/wiki/Sine_and_cosine_transforms#Fourier_inversion The first one in this column is simply the fourier integral applied over the above relation. Everything else is just a simplification into cosine

And this is even better...a whole justification of the connection http://en.wikipedia.org/wiki/Sine_and_cosine_transforms#Relation_with_complex_exponentials

-
Second link you provided is suplying link between time dependant and frequency dependant fourier. I have wave vector ependant and spatial dependant fourier. – 71GA Feb 25 at 23:53

1
Call it time or frequency or wave or space --- the formulas are the same, no? – Gerry Myerson Feb 26 at 0:17
{"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": 3, "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": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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": 0.8498948812484741, "perplexity_flag": "middle"}