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Line 16:
Line 16: − + + − + − <math>e^{yi}=1+yi-\frac{y^2}{2!}-i\frac{y^{3}}{3!}+\frac{y^4}{4!}+i\frac{y^5}{5!}-\ldots</math> − :<math>=(1-\frac{y^2}{2!}+\frac{y^4}{4!}-\ldots)+i(y-\frac{y^{3}}{3!}+\frac{y^5}{5!}-\ldots)</math> − +
→Extensions of the Exponential Function
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>
:<math>=1+yi-\frac{y^2}{2!}-i\frac{y^{3}}{3!}+\frac{y^4}{4!}+i\frac{y^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]]