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3.4 Neumann series

To develop a Neumann series of a general Fredholm equation, the function appearing in the integrand is expanded recursively. The first expansion yields

∫ { ∫ } f (Q0 ) = fi(Q0 )+ dQ1K (Q0,Q1 ) fi(Q1 )+ dQ2K (Q1, Q2)f (Q2 ) ∫ ∫ ∫ (3.20) = f◟i(◝Q◜0-)◞+ dQ1K (Q0,Q1 )fi(Q1 )+ dQ1 dQ2K (Q0,Q1 )K (Q1, Q2)f (Q2 ). f0 ◟---------◝◜1---------◞ ◟--------------2--◝3◜---n-------------◞ f f +f +⋅⋅⋅+f
The Neumann series is formed by an iterative application of the kernel K to the free term fi. The term fn corresponds to n applications of the kernel:
∑∞ f = fn; (3.21) n=0 { ∫ n-1 f n(Qn ) = dQnK (Qn -1,Qn )f (Qn ) if n > 0 (3.22) fi(Q0) if n = 0.
The evaluation of a Neumann series formally solves the Fredholm integral equation [124].

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