Find all solutions to the equation $e^z = i$

I know that there is an equation for finding the nth roots of a complex number, which easily done once you have the modulus and argument of the complex number in question. There would be n roots. But how do I know how many solutions there are to an exponential raised to a complex number?

Thanks for any help out there!

-

1
Use $e^{r\theta}=r\cdot cos (\theta) + r\cdot sin(\theta)i$ – you Mar 14 '12 at 0:21

4
@you: I am not sure that is quite correct – Henry Mar 14 '12 at 0:26
oops! It's too late for me to edit it now. I mean "$re^{i\theta}$" – you Mar 14 '12 at 4:14

2 Answers

$\displaystyle e^z=e^{i\frac{\pi}2+2k\pi i}$ so that $\ \displaystyle z=i\frac{\pi}2+2k\pi i\$ with $k\in \mathbb{Z}$

-

Thanks for your help! – mathlearner Mar 14 '12 at 5:31

For real $a$ and $b$ you have $$\exp(a+bi) = e^a \cos(b/2\pi) + e^a \sin(b/2\pi) i$$ so now set the right hand side equal to $i$ and solve the real and imaginary parts.

You get $e^a \cos(b/2\pi)=0$ but $e^a$ is positive so $b$ can only be of the form $(n+1/2)\pi$ for integer $n$, implying $\sin(b/2\pi)=\pm 1$, and you want $e^a \sin(b/2\pi)=1$ but $e^a$ is still positive so you must have $\sin(b/2\pi) = 1$ and so $e^a =1$, implying $b=(2n+1/2)\pi$ for integer $n$ and $a=0$, so the solution is $a+bi$ which is $$\left(2n+\frac{1}{2}\right)\pi i.$$

-

thanks for your help! – mathlearner Mar 14 '12 at 5:31
{"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": 23, "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": 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": 0.929267406463623, "perplexity_flag": "head"}