# Integrating without complex analysis

How can I evaluate this integral (without complex analysis)?

$$\int_{-\infty}^\infty\sinh [x(1-b)] \exp(iax) dx\qquad a, b\in \mathbb R$$

Thanks.

-

1
The way you have written this, it looks like $\sinh[a(1-b)]$ is just a constant which you can take outside of the integral. In this case the integral $\int_{-\infty}^{\infty} \exp(iax) dx$ does not converge. Maybe you meant to write something else. – Aleks Vlasev Nov 17 '11 at 9:03
Did you intend to have an $x$ somewhere in the $\sinh$? Yeah, what Aleks said ;-) – robjohn♦ Nov 17 '11 at 9:04
@AleksVlasev: Thanks, very well spotted! It's late and I'm losing it, anyway, I have edited it now. – peake Nov 17 '11 at 9:10
@robjohn: You are very right! :-) – peake Nov 17 '11 at 9:11

4
$\sinh(x)\to\pm\infty$ as $x\to\pm\infty$ so the integral doesn't converge, no complex analysis needed :-). – robjohn♦ Nov 17 '11 at 9:12
show 1 more comment

## 1 Answer

This is the Fourier transform of $\sinh$, which can be broken down as a sum of exponentials. Because the Fourier transform is a linear operator, your integral is a sum of Fourier transforms of exponentials, that is a sum of Lorentzian functions (as it's shown for instance here).

-
{"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": 1, "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": true}, "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": 0.9278088212013245, "perplexity_flag": "middle"}