For d-wave, $\Delta(k)=\Delta_{d} (\cos{(k_{x})}-\cos{(k_{y})})$

For s-wave, $\Delta(k)=\Delta_{0}$

For extended s-wave ,$\Delta(k)=\Delta_{s} (\cos{(k_{x})}+\cos{(k_{y})})$

How will be the nature of the density of states around $\omega=0$ for all the above three cases given $E_{k}=\sqrt{(\zeta(k)-\mu)^{2}+\Delta^{2}(k)}$ where $\zeta(k)=-2t(\cos{(k_{x})}+\cos{(k_{y})})$ (Square Lattice) and $\mu$ is the chemical potential?

Also, what will be the magnitude of the gap (in case there is a gap around $\omega=0$) ,the distance between the coherence peaks and the slope of the lines (in case linear around $\omega=0$)?

I understand that a Taylor series expansion is required, but how to go about it?


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