Is it Diffable? Gamma Stuff

Calculus Level pending

Let f f be the function defined on the reals by

f ( k ) = ( { ∫ 0 ∞ − cos ⁡ ( k π 2 ) sin ⁡ ( x 1 / k ) x 2 d x k > 0 ∨ k < − 1 ∫ 0 ∞ cos ⁡ ( k π 2 ) sin ⁡ ( x − 1 / k ) d x − 1 < k < 0 0 k = − 1 ∨ k = 0 ) f(k)=\left( \begin{array}{cc} \Bigg\{ & \begin{array}{cc} \int_0^{\infty } -\frac{\cos \left(\frac{k \pi }{2}\right) \sin \left(x^{1/k}\right)}{x^2} \, dx & k>0\lor k<-1 \\ \int_0^{\infty } \cos \left(\frac{k \pi }{2}\right) \sin \left(x^{-1/k}\right) \, dx & -1<k<0 \\ 0 & k=-1\lor k=0 \\ \end{array} \\ \end{array} \right)

Which of the following is true?

f f is continuous but f ′ f' is not f is not continuous f ′ f' is continuous but f ′ ′ f'' is not f is infinitely differentiable f ′ ′ f'' is continuous but f ( 3 ) f_{(3)} is not

Christopher Criscitiello by Christopher Criscitiello , 25, USA

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1 solution

f ( k ) = − π 2 Γ ( k ) f(k)=-\frac{\pi }{2 \Gamma (k)} , which (when extended to the complex plane) is holomorphic

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