Cosz

Cosz, typically written as cos z, denotes the cosine of a complex variable z. It is defined for all complex z by Euler's formula cos z = (e^{i z} + e^{-i z})/2. The function is entire, meaning it has no singularities on the complex plane.

Properties include that cos z is periodic with period 2π along the real axis and is even, satisfying cos(-z) = cos z. For z = x + i y, cos z can be written as cos z = cos x cosh y − i sin x sinh y, which shows how its magnitude grows with the imaginary part y.

Zeros of cos z occur at z = π/2 + kπ for integers k; all zeros are simple. In the imaginary direction, the function grows roughly like e^{|Im z|}. The function is of order 1 and is entire, with no poles.

Expansions and relationships include the Maclaurin series cos z = ∑_{n=0}^∞ (−1)^n z^{2n} /(2n)!. It also satisfies cos z = cosh(i z) and is related to the sine function by sin z = (e^{i z} − e^{−i z})/(2i). Its derivative is −sin z, and it satisfies the differential equation y'' + y = 0.

Usage and context: cos z is central in complex analysis and applied mathematics, used for solving differential equations, Fourier analysis, and contour integration. As the complex analogue of the real cosine, it extends familiar real-valued identities to the complex plane.