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Question Number 196322 by sniper237 last updated on 22/Aug/23
$$\:\:\:\:{f}^{\left(\mathrm{1}/\mathrm{2}\right)} \left({x}\right)=\:\frac{{d}}{{dx}}\left(\int_{\mathrm{0}} ^{{x}} \:\frac{{f}\left({x}−{t}\right)}{\:\sqrt{\pi{t}}}{dt}\right) \\ $$$${Prove}\:\:{that}\:\:\:\:\left({f}^{\left(\mathrm{1}/\mathrm{2}\right)} \right)^{\left(\mathrm{1}/\mathrm{2}\right)} =\:{f}\:'\:\:\:\: \\ $$$${At}\:\:{least}\:\:{for}\:\:{f}\:=\:\:\mathrm{1}\:\:{then}\:\:{f}\:=\:{x} \\ $$
Answered by witcher3 last updated on 22/Aug/23
$$\mathrm{Show}\:\mathrm{True}\:\mathrm{for}/\mathrm{All}\:\mathrm{f}\:\mathrm{derivable}\:? \\ $$$$\mathrm{or}\:\mathrm{just}\:\mathrm{f}=\mathrm{1}\:\mathrm{and}\:\mathrm{x}\:\mathrm{sorry}\:\mathrm{im}\:\mathrm{not}\:\mathrm{good}\:\mathrm{in}\:\mathrm{english} \\ $$$$ \\ $$