1
$\begingroup$

So I've been trying to get a good and intuitive feel for the extension of the sine and cosine functions into complex numbers (i.e. $\cos(z)$ where $z=a+bi$), and to do so I've naturally been looking at their definition using exponential functions

$$ \cos(x)=\frac{e^{ix}+e^{i(-x)}}{2} $$ $$ \sin(x)=\frac{e^{ix}-e^{i(-x)}}{2} $$

If we let $f(x)=e^{ix}$ and $g(x)=e^{i(-x)}$ , is there any way to imagine $f(x)$ or $g(x)$ on a purely real plane (or at least their influences that lead them to give "real" $\sin(x)$ and $\cos(x)$ graphs)? I understand that they form a circle when graphed in the complex plane, and to my understanding the cosine function is the real part of $e^{ix}$ while the sine function is the imaginary part when graphed against time, but I'm wondering if there is a more intuitive and less quantitative way to understand the relationship between complex exponential and trigonometric functions.

$\endgroup$
  • $\begingroup$ With gusto!${}$ $\endgroup$ – user98602 Jul 20 '15 at 5:38
  • $\begingroup$ I would start drawing $f(x)=e^{ix}$ as a 3D curve with axis Z related to the imaginary part of the image. $\endgroup$ – Masacroso Jul 20 '15 at 8:42
2
$\begingroup$

See : http://www.matematicasvisuales.com/english/html/complex/functions/cosine1.html

And the book "Visual Complex Analysis " by Tristan Needham.

| cite | improve this answer | |
$\endgroup$
-1
$\begingroup$

enter image description here![A picture is worth a thousand words and sin z does strange things to a face ]2

A picture is worth a thousand words and sin z does some strange things ~

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.