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Physics
The Gamma function
Factorials apply only to nonnegative integers but can be extended to complex numbers with positive real parts using the Gamma function, \(\Gamma(n) = \int_0^\infty e^{-x}x^{n-1}dx\).
Mathematics
The integral definition of the exponential function
The exponential function to the natural base, \(e^x\), can be defined in more than one way. One of them makes use of the integral \(\int_1^{e^x} \frac{dy}{y}= x\).
Physics
The heat equation
The heat equation, \(\frac{\partial \varnothing}{\partial t} = D \nabla^2 \varnothing\), uses the Laplacian of \(\varnothing\) to express diffusion through a medium.