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By Daniel A. Murray

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EXAMPLES. sjsin*} = {cos*}, s[f\ = {ηΓ-1} for *{β*} = {β*}+1, n > 1, s{t+l\ = {l}+l. Not only differentiate functions can be multiplied by the differential operator. The product sa will always be meaningful, whether a is a differentiate function, a non-differentiable one or an arbitrary operator. Thus if we regard multiplying by s as a generalization of differentiation, then in the domain of operators every continuous function (and even every integrable one) is differentiable (the result of the differentiation being an operator but not necessarily a function).

2) ansn+an^lsn-1+... + als+a0 = ßnsn + ßn_xsn-1 + . . 3) On^ßnj a n-\ = ßn-U ···> «1 = ß n «o^ßo- 38 III. 2) by Γ + 1, we obtain αηΖ + ... + α 0 Γ + 1 = βηΙ + ... + β0Γ+ϊ·, this equality means that a n + . . + a 0 - f =ßn + mmm + ß0— for 0 < t < oo. 3). Exercises. 1. Prove the formula l+*-M 2 -f-... + * n - 1 - ( s w - l ){*'}. 2. Verify that / Ä 4 + <$4 - / [ ( * - α ) 2 + α2]Γ(« + α ) 2 - ΰ 2 ] for a --—·—. § 24. Connections of the operator s with the exponential function. 1) s—a By the definition of convolution we have 1 (s-af _i (ί [e*\2 = { J V ^ V ^ T } = {** f dr\ = \~ — {p«t\{L·eÀ -αγ " I * 'tu A, — I Γ »«(*-*) ?..

The short-circuit current. In the preceding section we constantly assumed that 7(0) = 0 and Q(0) = 0. Now we shall assume 1(0) and Q(0) to be arbitrary and E = 0. This corresponds to the assumption that the terminals of the circuit are short-circuited at the instant t = 0 without any electromotive force being applied. Let us term the current which then flows through the circuit the short-circuit current and denote it by I. 1) ILS+R + J\Ï Csf =£1(0)- V(0) s where V(0) = —Q(0)/C is the potential difference on the condenser.

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