It's useful in physics for wave functions that might otherwise be cluttered with sines and cosines. It crops up from time to time in things that interest me. For example, the DE that defines a particular contour
[math]\frac{dz}{dt}=iz[/math] , solving for [math]z(t)[/math] where [math]t:0\to 1[/math]
Then
[math]z(t)={{z}_{0}}{{e}^{it}}[/math]
[math]{{z}_{0}}{{e}^{it}}={{z}_{0}}\left( \cos t+i\sin t \right)=\left( {{x}_{0}}\cos t-{{y}_{0}}\sin t \right)+i\left( {{y}_{0}}\cos t+{{x}_{0}}\sin t \right)[/math]
Comments (3)
[math display="block"]e^{i x} = \cos x + i \sin x, [/math]
Is it intended to be used for research purposes?
It's useful in physics for wave functions that might otherwise be cluttered with sines and cosines. It crops up from time to time in things that interest me. For example, the DE that defines a particular contour
[math]\frac{dz}{dt}=iz[/math] , solving for [math]z(t)[/math] where [math]t:0\to 1[/math]
Then
[math]z(t)={{z}_{0}}{{e}^{it}}[/math]
[math]{{z}_{0}}{{e}^{it}}={{z}_{0}}\left( \cos t+i\sin t \right)=\left( {{x}_{0}}\cos t-{{y}_{0}}\sin t \right)+i\left( {{y}_{0}}\cos t+{{x}_{0}}\sin t \right)[/math]