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Fredholm integral equations and the resolvent kernel

Consider the inhomogeneous Fredholm integral equation:

\[ H \psi(\mu) = \lambda \int_a^b \sigma(\mu, \mu') \psi(\mu') d\mu' + S(\mu) \]

The unknown to be solved for is \(\psi(\mu)\) where \(a \le \mu \le b\) is the independent variable, and the known function \(\sigma(\mu, \mu')\) is known as the kernel. This is an inhomogeneous equation because the known function \(S(\mu) \ne 0\).