Euler's Formula
Euler's Formula e^ix = cos x + i sin x We can generalize this to complex numbers: e^z = e^x (cos y + i sin y) where z = x + i y. In the applet, z is shown as a green dot on the complex plane. e^z is shown as a red dot. The unit circle is shown in gray. z goes from -π to π and back again, which causes e^z to go around the unit circle. e^z can be evaluated using a power series; the terms of this series are shown in alternating shades of blue, starting with 1 (the blue line from the origin to 1), and then z (the light blue line from 1 to 1+z), and then z^2/2, and so on.
Euler's Formula e^ix = cos x + i sin x We can generalize this to complex numbers: e^z = e^x (cos y + i sin y) where z = x + i y. In the applet, z is shown as a green dot on the complex plane. e^z is shown as a red dot. The unit circle is shown in gray. z goes from -π to π and back again, which causes e^z to go around the unit circle. e^z can be evaluated using a power series; the terms of this series are shown in alternating shades of blue, starting with 1 (the blue line from the origin to 1), and then z (the light blue line from 1 to 1+z), and then z^2/2, and so on.