Many-loop calculations: The uniqueness method and functional equations

In the framework of the calculation of many-loop Feynman integrals – the uniqueness method – functional equations are obtained for the coefficient functions of the diagrams. Solution of a functional equation leads to calculation of an $N$-shaped diagram, the last of the 5-loop diagrams of the $\varphi^4$ theory. The obtained result makes it possible to extend by an order the tables constructed previously for the calculation of many-loop integrals.