Let i = − 1 i=\sqrt{-1} i = − 1 e x = ∑ k = 0 ∞ x k k ! e^{x}=\sum_{k=0}^{\infty}\frac{x^{k}}{k!} e x = ∑ k = 0 ∞ k ! x k
e i x = ∑ k = 0 ∞ ( i x ) k k ! . e^{ix}=\sum_{k=0}^{\infty}\frac{\left(ix\right)^{k}}{k!}. e i x = k = 0 ∑ ∞ k ! ( i x ) k . Utilizing series expansion, this can be equivalently expressed this as
= 1 + i x + ( i x ) 2 2 ! + ( i x ) 3 3 ! + ( i x ) 4 4 ! + ( i x ) 5 5 ! + ( i x ) 6 6 ! + ( i x ) 7 7 ! + ( i x ) 8 8 ! + ⋯ = 1 + i x − x 2 2 ! − i x 3 3 ! + x 4 4 ! + i x 5 5 ! − x 6 6 ! − i x 7 7 ! + x 8 8 ! + ⋯ . \begin{array}{l}
=1+ix+\frac{\left(ix\right)^{2}}{2!}+\frac{\left(ix\right)^{3}}{3!}+\frac{\left(ix\right)^{4}}{4!}+\frac{\left(ix\right)^{5}}{5!}+\frac{\left(ix\right)^{6}}{6!}+\frac{\left(ix\right)^{7}}{7!}+\frac{\left(ix\right)^{8}}{8!}+\cdots\\
=1+ix-\frac{x^{2}}{2!}-\frac{ix^{3}}{3!}+\frac{x^{4}}{4!}+\frac{ix^{5}}{5!}-\frac{x^{6}}{6!}-\frac{ix^{7}}{7!}+\frac{x^{8}}{8!}+\cdots
\end{array}. = 1 + i x + 2 ! ( i x ) 2 + 3 ! ( i x ) 3 + 4 ! ( i x ) 4 + 5 ! ( i x ) 5 + 6 ! ( i x ) 6 + 7 ! ( i x ) 7 + 8 ! ( i x ) 8 + ⋯ = 1 + i x − 2 ! x 2 − 3 ! i x 3 + 4 ! x 4 + 5 ! i x 5 − 6 ! x 6 − 7 ! i x 7 + 8 ! x 8 + ⋯ . Because the series is absolutely convergent, the terms can be rearranged so to be expressed as
= ( 1 − x 2 2 ! + x 4 4 ! − x 6 6 ! + x 8 8 ! − ⋯ ) + i ( x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ ) = ∑ k = 0 ∞ ( − 1 ) k x 2 k ( 2 k ) ! + i ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 ( 2 k + 1 ) ! . \begin{array}{l}
=\left(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\cdots\right)+i\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots\right)\\
=\sum_{k=0}^{\infty}\frac{\left(-1\right)^k x^{2k}}{\left(2k\right)!}+i\sum_{k=0}^{\infty}\frac{\left(-1\right)^k x^{2k+1}}{\left(2k+1\right)!}
\end{array}. = ( 1 − 2 ! x 2 + 4 ! x 4 − 6 ! x 6 + 8 ! x 8 − ⋯ ) + i ( x − 3 ! x 3 + 5 ! x 5 − 7 ! x 7 + ⋯ ) = ∑ k = 0 ∞ ( 2 k ) ! ( − 1 ) k x 2 k + i ∑ k = 0 ∞ ( 2 k + 1 ) ! ( − 1 ) k x 2 k + 1 . By recognizing the Maclaurin series for cos x \cos x cos x sin x \sin x sin x
e i x = cos x + i sin x . e^{ix}=\cos x+i\sin x. e i x = cos x + i sin x . This is called Euler’s Formula,e x e^x e x