Editing Transfer function (section)

=== Direct derivation from differential equations ===
A [[linear differential equation]] with constant coefficients

:<math> L[u] = \frac{d^nu}{dt^n} + a_1\frac{d^{n-1}u}{dt^{n-1}} + \dotsb + a_{n-1}\frac{du}{dt} + a_nu = r(t) </math>

where ''u'' and ''r'' are suitably smooth functions of ''t'', has ''L'' as the operator defined on the relevant function space that transforms ''u'' into ''r''. That kind of equation can be used to constrain the output function ''u'' in terms of the ''forcing'' function ''r''. The transfer function can be used to define an operator <math>F[r] = u </math> that serves as a right inverse of ''L'', meaning that  <math>L[F[r]] = r</math>.

Solutions of the homogeneous [[Linear differential equation#Homogeneous equations with constant coefficients|constant-coefficient differential equation]] <math>L[u] = 0</math> can be found by trying <math>u = e^{\lambda t}</math>. That substitution yields the [[Characteristic equation (calculus)|characteristic polynomial]]

:<math> p_L(\lambda) = \lambda^n + a_1\lambda^{n-1} + \dotsb + a_{n-1}\lambda + a_n\,</math>

The inhomogeneous case can be easily solved if the input function ''r'' is also of the form <math>r(t) = e^{s t}</math>. By substituting <math>u = H(s)e^{s t}</math>, <math>L[H(s) e^{s t}] = e^{s t}</math> if we define

:<math>H(s) = \frac{1}{p_L(s)} \qquad\text{wherever }\quad p_L(s) \neq 0.</math>

Other definitions of the transfer function are used, for example <math>1/p_L(ik) .</math><ref>{{cite book |title= Ordinary differential equations|last= Birkhoff |first= Garrett|author2=Rota, Gian-Carlo |year=1978|publisher=John Wiley & Sons |location= New York|isbn= 978-0-471-05224-1}}{{page needed|date=April 2013}}</ref>