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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9bd4c8310c811586128145e30f9063dc227b477)
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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d207499641004287d496c112398531fd0845aa9)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33699519125da5722b6f436347de3b1e0aaef3a4)
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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/614d2b820bfe8a8fbc1e959e87c7aa6b612ed99a)
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)}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25e44fb1e62b11b2912ca69dededcd89ea3b7665)
- 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