Exercise 15.4.15

Proof. To know the first coefficients of the power series expansion of $\phi (\mathit{\beta z})$, it is sufficient to do an asymptotic expansion in the neighborhood of $0$ up to degree 9.

By (15.54),

The second line gives

Since ${(1+w)}^5=1+5w+o(w)$, where $w=c_1{z}^4+o(z^4)\to 0$ when $z\to 0$, and $w\sim c_1{z}^4$, thus $o(w)=o(z^4)$, we obtain

so that

Moreover

Therefore

thus the power series expansion of $\phi (\mathit{\beta z})$ begins with

This gives (15.58). □