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How are exponential and trigonometric functions extended to complex numbers?

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transformation of a circular region into an approximately rectangular region — analysis \
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr². As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr²—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

In mathematics, exponential and trigonometric functions are extended to the complex plane by utilizing their Taylor series expansions. By replacing the real variable $x$ with a complex variable $z$ , these functions are defined for all complex numbers.123

The exponential function is defined by the power series:
$e^{z} = 1 + z + \frac{z^{2}}{2!} + \frac{z^{3}}{3!} + \dots$ 3

Trigonometric functions are then related to the exponential function through Euler’s formula, which states that $e^{i θ} = \cos (θ) + i \sin (θ)$ . By applying this relationship, the sine and cosine functions for a complex variable $z$ are expressed as:43
$\cos (z) = \frac{e^{i z} + e^{- i z}}{2}$
$\sin (z) = \frac{e^{i z} - e^{- i z}}{2 i}$ 43

This approach demonstrates that trigonometric functions are fundamentally linked to the exponential function, allowing for a unified treatment of these transcendental functions within complex analysis.32