Logaritmoidun

Logaritmoidun is a hypothetical mathematical function defined for positive real numbers by the formula logaritmoidun(x) = ln(x) + arctan(x). The name suggests a fusion of logarithmic growth with the saturating behavior of the arctangent, and the construction is used as a teaching and exploration tool in the study of composite functions.

Analytical properties. For x > 0, logaritmoidun is differentiable with derivative logaritmoidun'(x) = 1/x + 1/(1 + x^2). This derivative is strictly positive on the positive real axis, so logaritmoidun is strictly increasing. The second derivative is logaritmoidun''(x) = −1/x^2 − 2x/(1 + x^2)^2, which is negative for x > 0, making logaritmoidun a concave function on its domain. As x approaches 0 from the right, logaritmoidun(x) tends to −∞, while as x grows large, logaritmoidun(x) behaves like ln(x) with a finite additive term arctan(x) approaching π/2; consequently, logaritmoidun is bijective from (0, ∞) onto the real numbers and possesses a numerically computable inverse.

Examples. Values illustrate its behavior: logaritmoidun(1) = ln(1) + arctan(1) = π/4 ≈ 0.7854; logaritmoidun(2) ≈ 0.6931 + 1.1071 ≈ 1.8002; logaritmoidun(0.5) ≈ −0.6931 + 0.4636 ≈ −0.2295.

Generalizations and uses. Variations include logaritmoidun_k(x) = ln(x) + arctan(kx) for k > 0, or replacing arctan with other saturating functions to study how different limits affect shape and invertibility. Applications are primarily instructional, illustrating how combining growth and saturation influences monotonicity, concavity, and invertibility.

See also: natural logarithm, arctangent, concave functions, inverse functions.