ipi2ln10

ipi2ln10 is a symbolic expression that denotes the product i × π × 2 × ln(10). In compact form it is often written as 2 i π ln(10) or iπ2ln10. It is not a standard named constant in mainstream mathematics, but rather a simple complex product of well-known constants: the imaginary unit i, π, and the natural logarithm of 10.

Numerically, 2 π ln(10) evaluates to approximately 14.46757, so ipi2ln10 equals approximately 14.46757 i. This makes ipi2ln10 a purely imaginary number with magnitude about 14.46757 and an argument of π/2 (modulo 2π). The quantity is frequently encountered in contexts involving complex exponentials, since multiplying by i places a real quantity on the imaginary axis.

In exponential form, e^{ipi2ln10} can be written as e^{i(2π ln(10))} = cos(2π ln(10)) + i sin(2π ln(10)). Because ln(10) is irrational with respect to π, the angle 2π ln(10) is not a simple rational multiple of π, which has implications in various theoretical considerations of oscillatory behavior.

Context and caveats: ipi2ln10 is not a standardized constant, and the lack of explicit multiplication signs or parentheses can lead to misinterpretation. When used, it is advisable to write the expression clearly as i(2π ln(10)) or 2 i π ln(10) to avoid ambiguity.

See also: imaginary unit i, π, natural logarithm, complex exponentials, Euler's formula.