ExpFxy

ExpFxy is a shorthand notation used in mathematical contexts to denote the exponential of a two-variable function F, written as exp(F(x,y)) or e^{F(x,y)}. It is not a standardized object with a single definition, but rather a convenient way to reference the quantity when F: R^2 -> R is real-valued and the exponential appears in equations.

Notation and generalization

If F is real-valued, ExpFxy means e^{F(x,y)}. If F takes complex values, the exponential is still defined with the complex exponential. When F is matrix-valued, the matrix exponential exp(F(x,y)) is used, typically written as expm(F(x,y)) rather than ExpFxy, since matrix exponentials involve different properties.

Properties

For real-valued F, ExpFxy > 0 for all (x,y) in the domain. Exponential rules apply: ExpFxy · ExpGxy = Exp(F+G) and ExpFxy is differentiable wherever F is differentiable. Partial derivatives satisfy ∂/∂x ExpFxy = ExpFxy · ∂F/∂x and similarly for y. If F is smooth, higher-order derivatives follow by the chain rule.

Examples

If F(x,y) = ax + by + c, then ExpFxy = e^{ax+by+c} = e^{c} e^{ax} e^{by}. If F(x,y) = -((x-μ)^2+(y-ν)^2)/(2σ^2), ExpFxy yields a Gaussian-shaped function up to a constant factor.

Applications

ExpFxy appears in statistics and physics, for example in Boltzmann weights e^{-E(x)} and in the exponential family of distributions, where the exponent governs likelihoods or energy terms. It also arises in solutions to PDEs and in energy-based models.

Computational notes

Numerical evaluation must manage overflow and underflow; stable techniques include log-exp transforms and the log-sum-exp trick when combining multiple exponentials. For complex F, care with branches of the complex exponential is required.