Find the solution \(\displaystyle\lim_{{{t}\to{0}}}{\frac{{{d}^{{{k}-{1}}}}}{{{\left.{d}{t}\right.}^{{{k}-{1}}}}}}{\left({\frac{{{t}\sqrt{{{1}+{t}}}}}{{{\log{\sqrt{{{1}+{t}}}}}}}}\right)}^{{k}}={2}{\left({k}+{2}\right)}^{{{k}-{1}}}\)

Justine White 2022-04-07 Answered
Find the solution
limt0dk1dtk1(t1+tlog1+t)k=2(k+2)k1
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Answers (1)

superpms01wks1
Answered 2022-04-08 Author has 13 answers
Complex analysis comes to the rescue. Using Cauchy differentiation formula:
fn1(0)=(n1)!2πif(z)zndz
Now
limt0dk1dtk1(t1+tlog1+t)k=(k1)!2πi(t1+tlog1+t)kdttk
(k1)!2πi(1+tlog1+t)kdt
Now performing the change of variable t=eu1
limt0dk1dtk1(t1+tlog1+t)k=(k1)!2πi(exp(u2)u2)keudu
=2k[(k1)!2πiexp(u(k2+1)}{uk}du
=2klimu0dk1duk1eu(k2+1)=2k(k2+1)k1=2(k+2)k1
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