Exercise 6, § 2

Functions of a Real Variable, IV, 205

Let $f(X)$ be a polynomial of degree $n$ with complex coefficients, and let

\frac{1}{f(X)} = \sum_{j=1}^{q} \sum_{h=1}^{n_j} \frac{\lambda_{jh}}{(X - r_j)^h}

be the decomposition of the rational fraction $1/f(X)$ into simple elements (Alg., VII. 7, n 2). Show that, for any regulated function $b$ , the function

t \mapsto \sum_{j=1}^{q} \sum_{h=1}^{n_j} \lambda_{jh} \int_{t_0}^{t} \frac{(t-s)^{h-1}}{(h-1)!} e^{l_j(t-s)} b(s) ds

is an integral of the equation $f(D)x = b(t)$ .

Solution

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