Carleman's equation

In mathematics, Carleman's equation is a Fredholm integral equation of the first kind with a logarithmic kernel. Its solution was first given by Torsten Carleman in 1922. The equation is

${\displaystyle \int _{a}^{b}\ln |x-t|\,y(t)\,dt=f(x)}$

The solution for b − a ≠ 4 is

${\displaystyle y(x)={\frac {1}{\pi ^{2}{\sqrt {(x-a)(b-x)}}}}\left[\int _{a}^{b}{\frac {{\sqrt {(t-a)(b-t)}}f'_{t}(t)\,dt}{t-x}}+{\frac {1}{\ln \left[{\frac {1}{4}}(b-a)\right]}}\int _{a}^{b}{\frac {f(t)\,dt}{\sqrt {(t-a)(b-t)}}}\right]}$

If b − a = 4 then the equation is solvable only if the following condition is satisfied

${\displaystyle \int _{a}^{b}{\frac {f(t)\,dt}{\sqrt {(t-a)(b-t)}}}=0}$

In this case the solution has the form

${\displaystyle y(x)={\frac {1}{\pi ^{2}{\sqrt {(x-a)(b-x)}}}}\left[\int _{a}^{b}{\frac {{\sqrt {(t-a)(b-t)}}f'_{t}(t)\,dt}{t-x}}+C\right]}$

where C is an arbitrary constant.

For the special case f(t) = 1 (in which case it is necessary to have b − a ≠ 4), useful in some applications, we get

${\displaystyle y(x)={\frac {1}{\pi \ln \left[{\frac {1}{4}}(b-a)\right]}}{\frac {1}{\sqrt {(x-a)(b-x)}}}}$

References

• CARLEMAN, T. (1922) Uber die Abelsche Integralgleichung mit konstanten Integrationsgrenzen. Math. Z., 15, 111–120
• Gakhov, F. D., Boundary Value Problems [in Russian], Nauka, Moscow, 1977
• A.D. Polyanin and A.V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4