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

$\underset{a}{\overset{b}{\int }}\ln |x-t|y(t)dt=f(x)$

The solution for b − a ≠ 4 is

$y(x)=\frac{1}{{\pi }^2\sqrt{(x-a)(b-x)}}[\underset{a}{\overset{b}{\int }}\frac{\sqrt{(t-a)(b-t)}f{\text{'}}_t(t)dt}{t-x}+\frac{1}{\ln [\frac{1}{4}(b-a)]}\underset{a}{\overset{b}{\int }}\frac{f(t)dt}{\sqrt{(t-a)(b-t)}}]$

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

$\underset{a}{\overset{b}{\int }}\frac{f(t)dt}{\sqrt{(t-a)(b-t)}}=0$

In this case the solution has the form

$y(x)=\frac{1}{{\pi }^2\sqrt{(x-a)(b-x)}}[\underset{a}{\overset{b}{\int }}\frac{\sqrt{(t-a)(b-t)}f{\text{'}}_t(t)dt}{t-x}+C]$

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

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

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. Template:ISBN