Green's function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a Green's function (or Green function) is the <a href="/facts/Impulse_response/l4vaE5er">impulse response</a> of an <a href="/facts/Inhomogeneous_ordinary_differential_equation/ygKGQ8kD">inhomogeneous</a> linear <a href="/facts/Differential_operator/Nr15J0IE">differential operator</a> defined on a domain with specified initial conditions or boundary conditions.
This means that if 
 
 
 
 L
 
 
 {\displaystyle L}
 
 is a linear differential operator, then

<ul><li>the Green's function 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is the solution of the equation 
 
 
 
 L
 G
 =
 δ
 
 
 {\displaystyle LG=\delta }
 
, where 
 
 
 
 δ
 
 
 {\displaystyle \delta }
 
 is <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac's delta function</a>;</li>
<li>the solution of the initial-value problem 
 
 
 
 L
 y
 =
 f
 
 
 {\displaystyle Ly=f}
 
 is the <a href="/facts/Convolution/PVPrdz9J">convolution</a> (
 
 
 
 G
 ∗
 f
 
 
 {\displaystyle G\ast f}
 
).</li></ul>
Through the <a href="/facts/Superposition_principle/GKGs9C92">superposition principle</a>, given a <a href="/facts/Linear_differential_equation/nLEbIIVf">linear ordinary differential equation</a> (ODE), 
 
 
 
 L
 y
 =
 f
 
 
 {\displaystyle Ly=f}
 
, one can first solve 
 
 
 
 L
 G
 =
 
 δ
 
 s
 
 
 
 
 {\displaystyle LG=\delta _{s}}
 
, for each s, and realizing that, since the source is a sum of <a href="/facts/Delta_function/V8PjRP8T">delta functions</a>, the solution is a sum of Green's functions as well, by linearity of L.
Green's functions are named after the British <a href="/facts/Mathematician/eaFSR8Fx">mathematician</a> <a href="/facts/George_Green_(mathematician)/89DMr6wz">George Green</a>, who first developed the concept in the 1820s. In the modern study of linear <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equations</a>, Green's functions are studied largely from the point of view of <a href="/facts/Fundamental_solution/hdu8vXJa">fundamental solutions</a> instead.
Under <a href="/facts/Green%2527s_function_(many-body_theory)/1v3ethbc">many-body theory</a>, the term is also used in <a href="/facts/Physics/a7aH2y08">physics</a>, specifically in <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a>, <a href="/facts/Aerodynamics/Cn6Lnl3H">aerodynamics</a>, <a href="/facts/Aeroacoustics/FJN943nB">aeroacoustics</a>, <a href="/facts/Electrodynamics/oAkmALHT">electrodynamics</a>, <a href="/facts/Seismology/Lony1LYX">seismology</a> and <a href="/facts/Statistical_field_theory/EeYlVGoB">statistical field theory</a>, to refer to various types of <a href="/facts/Correlation_function_(quantum_field_theory)/zH7RInAw">correlation functions</a>, even those that do not fit the mathematical definition. In quantum field theory, Green's functions take the roles of <a href="/facts/Propagator/FER8tNKP">propagators</a>.

Green's function open-in-new

Green's function