Summary: In this paper, we state a theorem for the inverse Laplace transform of functions involving conjugate branch points on imaginary axis. We get two equivalent integral representations for this inversion in terms of the Fourier sine and cosine transforms. Also, as applications of this theorem we obtain the solutions of dual integral equations with kernels of the Struve and Bessel functions and show an integral representation for the Gregory-Nörlund numbers. Moreover, new representations of the powers of the Airy functions are given in terms of the fractional integrals of order \(\frac{1}{2}\).