📂Complex Anaylsis

Relationship Between Trigonometric and Exponential Functions in Complex Analysis

Theorem ¹

As complex functions, the sine and cosine functions $\sin , \cos : \mathbb{C} \to \mathbb{C}$ are as follows. $$ \sin z = { {e^{iz} - e^{-iz}} \over 2 i } \\ \cos z = { {e^{iz} + e^{-iz}} \over 2 } $$

Explanation

In fact, rather than a theorem, one may as well regard this as a definition. The point is to show that defining them this way causes no conflict with previously established theorems. The proof, too, is nothing more than rearranging what we already knew from Euler’s formula in terms of the trigonometric functions.

Proof

By Euler’s formula $\displaystyle { e }^{ ix }= \cos x + i \sin x$, $$ \begin{cases} { e }^{ iz }= \cos z + i \sin z \\ { e }^{ -iz }= \cos z - i \sin z \end{cases} $$ Solving these for the trigonometric functions gives $$ \sin z = { {e^{iz} - e^{-iz}} \over 2 i } \\ \cos z = { {e^{iz} + e^{-iz}} \over 2 } $$

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