Changes

Consider the exponential of [[imaginary number]] <math>yi</math>,

Consider the exponential of [[imaginary number]] <math>yi</math>,

<math>e^{yi}=1+yi+\frac{(yi)^2}{2!}+\frac{(yi)^3}{3!}+\frac{(yi)^4}{4!}+\frac{(yi)^5}{5!}+\ldots</math>,

<math>e^{yi}=1+yi+\frac{(yi)^2}{2!}+\frac{(yi)^3}{3!}+\frac{(yi)^4}{4!}+\frac{(yi)^5}{5!}+\ldots</math>

:as <math>i^{2}=-1</math>

:<math>=(1-\frac{y^2}{2!}+\frac{y^4}{4!}-\ldots)+i(y-\frac{y^{3}}{3!}+\frac{y^5}{5!}-\ldots)</math>

:<math>=\cos y+i\sin y</math>

:<math>=\cos y+i\sin y</math>

By the power laws then,

By the power laws then all [[complex numbers] have an exponential,

<math>e^{x+yi}=e^{x}(\cos y+i\sin y)</math>.

<math>e^{x+yi}=e^{x}(\cos y+i\sin y)</math>.

[[category:mathematics]]

[[category:mathematics]]