How to use steepest descent method to approximate $int_0^{1}s^{1/4+i b x}e^{sx}ds$ as $xto+infty$?
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Let $0< bleqslant 1$. I am interested in using the steepest descent method to calculate the asymptotic approximation (as $xto+infty$) to the following integral that is related to function ${_1}F_1(1,9/4+ibx,-x)$: $$J(b,x)=int_0^{1}s^{1/4+i b x}e^{sx}ds=int_0^{1}s^{1/4}exp(x(s+iblog s))dstag{1}$$ Let $rho(z)=z+iblog z$. From several excellent solutions in the posts here and here, I know that we suppose to find contour segments such that $Im(rho(z))$=const. I was not able to find explicit solutions when I set $z=u+iv$ or $z=re^{iphi}$. Also $rho'(z)=1+ib/z$ so the saddle point is at $z_0=-ib$. And $rho^{"}(z_0)=i/b$. Thanks for any suggestions. Update Set $z=re^{iphi}$ in $$rho(z)=z+iblog z,$$ we obtain $$Rerho(z)=rcosphi-bphi,quadImrho(z)=rsinphi+blog r$$ To make $Imrho(z