This question is from the paper, The Analysis of Elliptic Families II. Dirac Operators, Eta Invariants, and the Holonomy Theorem, Commun. Math. Phys. 107, 103-163 (1986)---Proposition 2.8.

Suppose that $Tr[D^uexp(-tD^2)]=\frac{C_{-n/2}}{t^{n/2}}+\cdots+ \frac{C_{-1/2}}{t^{1/2}}+O(t^{1/2})$ as $t\to0$. Here $D$ denotes the Dirac operator and $D^u$ denotes the $u$-derivative of $D$. (I think one could ignore the exact definition here.)

And, we have that $$\Gamma(\frac{s+1}2)\eta(s)=-s\int^\infty_0t^{\frac{s-1}2}Tr[D^uexp(-tD^2)] dt.$$

Q By the asymptotic formula of the "heat kernel", how could we deduce that $$\eta(0)=-2C_{-1/2}/\sqrt\pi.$$

PS: What I do not understand:

• 1. Why the term could vanish under the integral?
• 2. Since the integral is from zero to infinity, is it safe to use the asymptotic formula near $0$?