lnsin

lnsin is the natural logarithm of the sine function, defined for real numbers x by f(x) = ln(sin x). It is real-valued on the portions of the real line where sin x is positive.

Domain and range: The sine is positive on the intervals (2kπ, (2k+1)π) for integers k, so the domain of f is the union of these intervals. Since sin x ∈ (0, 1], we have ln(sin x) ∈ (-∞, 0], attaining 0 at points where sin x = 1 (x = π/2 + 2kπ) and tending to -∞ as x approaches 0 or π within an interval.

Key properties: The derivative is f′(x) = cot x, and the second derivative is f′′(x) = -csc^2 x. Thus f is smooth on its domain and concave on each interval where sin x > 0. The function is periodic with period 2π, though it is only defined on the half-period intervals where sin x > 0.

Examples and notable results: A classical integral involving lnsin is ∫_0^π ln(sin x) dx = -π ln 2, a standard result in analysis. The graph of lnsin repeats its pattern on each interval (2kπ, (2k+1)π), with a maximum of 0 at x = π/2 + 2kπ and vertical-like behavior (going to -∞) near the interval endpoints.

Extensions and related notions: For complex arguments, one may consider f(z) = ln(sin z) using the complex logarithm, which is multi-valued and requires a choice of branch along with the zeros of sin z. Related real functions include the sine, the natural logarithm, and the cotangent via its derivative.