# Nature of density of states for d-wave, s-wave and extended s-wave pairing symmetries

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?