In Appendix A of this paper, it is claimed that the asymptotic behaviour of

$$\phi_1(y,\lambda)=\frac{1}{\Gamma(\frac{1-\lambda}{2})}\int_0^\infty dt~e^{-t}\cos(2y\sqrt{t})t^{-\frac{1+\lambda}{2}},$$

for $\lambda_2=\text{Im}(\lambda)\to+\infty$ (where $\lambda=\lambda_1+i\lambda_2$), is

$$\phi_1(y,\lambda_1+\lambda_2)\sim\cosh[y\sqrt{\lambda_2-i\lambda_1}(1+i)]\exp(-y^2/2) .$$

I have been trying various asymptotic methods to arrive at this result, such as the method of stationary phase, without success. How might one derive this result?

In Appendix A of this paper, it is claimed that the asymptotic behaviour of

$$\phi_1(y,\lambda)=\frac{1}{\Gamma(\frac{1-\lambda}{2})}\int_0^\infty dt~e^{-t}\cos(2y\sqrt{t})t^{-\frac{1+\lambda}{2}},$$

for $\lambda_2=\text{Im}(\lambda)\to+\infty$ (where $\lambda=\lambda_1+i\lambda_2$), is

$$\phi_1(y,\lambda_1+i\lambda_2)\sim\cosh[y\sqrt{\lambda_2-i\lambda_1}(1+i)]\exp(-y^2/2) .$$

I have been trying various asymptotic methods to arrive at this result, such as the method of stationary phase, without success. How might one derive this result?