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Suppose an $s$-stage Runge-Kutta method has order $2s-1$, an invertible matrix $A$, and $b^T{A}^{-1}1=1$. Its stability function is $R(z)=1+z{b}^T(I-zA{)}^{-1}1$. The adjugate matrix formula gives a rational function with denominator degree $s$ and numerator degree at most $s$; its zero limit at infinity lowers the latter degree to at most $s-1$. The order of a numerical method gives $R(z)-e^z=O(z^{2s})$, identifying the $[s-1/s]$ Padé approximant. Uniqueness follows because the cross-multiplied difference of two candidates has degree at most $2s-1$ but a zero of order at least $2s$, so is identically zero. The A-stability of near-diagonal exponential Padé approximants then proves A-stability, and the zero limit proves L-stability. These conditions identify the stability function; they do not alone claim uniqueness of all entries of the Runge-Kutta method tableau.

Apply the Runge-Kutta method to the Dahlquist test equation and put $z=h\lambda$. Its stability function isSince $A$ is an invertible matrix, $Q$ has polynomial degree $s$. The adjugate matrix formula gives $\mathrm{deg}P\le s$, but the additional condition givesso actually $\mathrm{deg}P\le s-1$. The specified order of a numerical method implies $R(z)=e^z+O(z^{2s})$ near zero. Therefore the Padé identification of a Radau stability function makes $R$ the $[s-1/s]$ Padé approximant to the exponential function: its numerator and denominator have the required degrees, and its Taylor series agrees through degree $2s-1$.

In the permitted A-stability of near-diagonal exponential Padé approximants, take $m=s-1$, $n=s$. Then $n-2\le m\le n$, so the scheme is A-stable. The same stability function also tends to zero at infinity, giving L-stability.