# Abel integral equation

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The integral equation

 (1)

to which the solution of the Abel problem is reduced. The generalized Abel integral equation is the equation

 (2)

where and are known constants, is a known function and is the unknown function. The expression is called the kernel of the Abel integral equation, or Abel kernel. An Abel integral equation belongs to the class of Volterra equations of the first kind (cf. Volterra equation). The equation

 (3)

is called Abel's integral equation with fixed limits.

If is a continuously-differentiable function, then the Abel integral equation (2) has a unique continuous solution given by the formula

 (4)

or

 (5)

which is the same thing. Formula (5) also gives the solution of an Abel integral equation (2) under more general assumptions [3], [4]. Thus, it has been shown [3] that if is absolutely continuous on the interval , the Abel integral equation (2) has the unique solution in the class of Lebesgue-integrable functions given by formula (5). For the solution of the Abel integral equation (3) see [2]; see also [6].

#### References

 [1] M. Bôcher, "On the regions of convergence of power-series which represent two-dimensional harmonic functions" Trans. Amer. Math. Soc. , 10 (1909) pp. 271–278 [2] T. Carleman, "Ueber die Abelsche Integralgleichung mit konstanten Integrationsgrenzen" Math. Z. , 15 (1922) pp. 111–120 [3] L. Tonelli, "Su un problema di Abel" Math. Ann. , 99 (1928) pp. 183–199 [4] J.D. Tamarkin, "On integrable solutions of Abel's integral equation" Ann. of Math. (2) , 31 (1930) pp. 219–229 [5] S.G. Mikhlin, "Linear integral equations" , Hindushtan Publ. Comp. , Delhi (1960) (Translated from Russian) [6] F.D. Gakhov, "Boundary value problems" , Pergamon (1966) (Translated from Russian)