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Let f(x)=e^{ax}. In numerical analysis this function is important because derivatives, Taylor expansions and finite differences adopt closed forms. Differentiation gives f'(x)=a e^{ax}, and by induction f^{(n)}(x)=a^n e^{ax}. Hence Taylor series about x0 with step h is e^{a(x0+h)}=e^{ax0}\sum\_{k=0}^{\infty}\frac{(a h)^k}{k!}, which converges for all h, so truncation error can be expressed exactly using remainder formulas. The shift identity f(x+h)=e^{a h} f(x) implies ______ ____ ______ _______ ___ ________ __________ _____.
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