| Line 5: | Line 5: | ||
Euler's formula | Euler's formula | ||
| − | <math> e^{jx} = \cos x + | + | <math> e^{jx} = \cos x + j \sin x \,\! </math> |
| − | <math> \cos x = \mathrm{Re}\{e^{ | + | <math> \cos x = \mathrm{Re}\{e^{jx}\} ={e^{jx} + e^{-jx} \over 2}</math> |
| − | <math> \sin x = \mathrm{Im}\{e^{ | + | <math> \sin x = \mathrm{Im}\{e^{jx}\} ={e^{jx} - e^{-jx} \over 2i}. </math> |
<math> \cos(x) = {e^{-jx} + e^{jx} \over 2}</math> | <math> \cos(x) = {e^{-jx} + e^{jx} \over 2}</math> | ||
<math> \sin(x) = {e^{-jx} - e^{jx} \over 2j} </math> | <math> \sin(x) = {e^{-jx} - e^{jx} \over 2j} </math> | ||
Latest revision as of 19:40, 22 July 2009
Euler's identity
$ e^{j \pi} + 1 = 0, \,\! $
Euler's formula
$ e^{jx} = \cos x + j \sin x \,\! $
$ \cos x = \mathrm{Re}\{e^{jx}\} ={e^{jx} + e^{-jx} \over 2} $
$ \sin x = \mathrm{Im}\{e^{jx}\} ={e^{jx} - e^{-jx} \over 2i}. $
$ \cos(x) = {e^{-jx} + e^{jx} \over 2} $
$ \sin(x) = {e^{-jx} - e^{jx} \over 2j} $
