10.4.30 Laplace Transform

Condition: find (continuous on \ ([0;+\ infty)) \) the function of theorine, if its image \ (f (p) \) is: 1) \ (f (p) = \ int_ {p}^{+\ infty} \ frac {d q} {q \ left (q^{2} +1 \ right)} \), 2) \ (f (p) = \ sum_ {k = 0}^{\ infty} \ frac {1} {p^{k+1} \), 3) \ ( F (p) = \ sum_ {k = 0}^{\ infty} \ frac {1} {(2 p)^{k+1}} \).