From: Mathematical analysis of a generalized epidemic model with nonlinear incidence function
Original function f(x) | Transformed function F(k) |
---|---|
f(x) = g(x) ± h(x) | F(k) = G(k) ± H(k) |
f(x) = cg(x) | F(k) = cG(k), where c is a constant |
\( f(x)=\frac{dg(x)}{dx} \) | F(k) = (k + 1)G(k + 2) |
\( f(x)=\frac{d^mg(x)}{dx^m} \) | F(k) = (k + 1)(k + 2)…(k + m)G(k + m) |
f(x) = 1 | F(k) = δ(k) |
f(x) = x | F(k) = δ(k − 1) |
f(x) = xm | \( F(k)=\delta \left(k-m\right)=\Big\{{}_{0,\kern0.5em if\ k\ne m}^{1,\kern0.5em if\ k=m} \) |
f(x) = g(x)h(x) | \( F(k)={\sum}_{m=0}^kH(m)G\left(k-m\right) \) |
f(x) = emx | \( F(k)=\frac{m^k}{k!} \) |
f(x) = (1 + x)m | \( F(k)=\frac{m\left(m-1\right)\left(m-2\right)\dots \left(m-k+1\right)}{k!} \) |