exponimus

Exponimus is a term used to denote the generalized exponentiation operator. In its standard interpretation, exponimus(a, b) equals a raised to the power b, typically written as a^b, with the domain chosen to reflect conventional exponent rules.

Notation and domains: For real numbers, exponimus is defined when a > 0 for any real b; with negative bases it is defined only for integer exponents or certain rational exponents. In the complex plane, exponimus uses a^b = exp(b Log a), yielding multiple values; the principal value is commonly used unless stated otherwise.

Variants and extensions: In modular arithmetic, exponimus is defined as a^b mod n and is central to many cryptographic protocols. In symbolic computation, exponimus aids algebraic manipulation and exact exponent handling, enabling simplification and transformation under defined conditions.

Properties: Exponimus obeys standard laws when defined, such as a^{b+c} = a^b a^c and (a^b)^c = a^{bc}, with a^0 = 1 for a ≠ 0. The case a = 0 is restricted to positive exponents; 0^0 is indeterminate in many contexts.

Computational aspects and applications: Efficient calculation relies on exponentiation by squaring for real and integer exponents, and on modular reduction for modular cases. Exponimus appears in cryptography, numerical analysis, and computer algebra systems, as well as in models of growth and decay in the sciences.

Origin and terminology: Exponimus is a neologism used to discuss exponentiation in didactic and theoretical contexts; it is not a separate mathematical object beyond the standard exponentiation operator.