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

: $\int_a^b \ln, x-t, \, y(t) \, dt = f(x)$

The solution for ''b'' − ''a'' ≠ 4 is

:
y(x)
=
\frac
\left \int_a^b \frac
+\frac \int_a^b \frac
\right

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

: $\int_a^b \frac = 0$

In this case the solution has the form

:
y(x)
=
\frac
\left \int_a^b \frac
+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
: $y(x) = \frac \frac$

References

* CARLEMAN, T. (1922) Uber die Abelsche Integralgleichung mit konstanten Integrationsgrenzen. Math. Z., 15, 111–120
* Gakhov, F. D., Boundary Value Problems n 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

Fredholm theory
Integral equations