Leading term of an asymptotic expansion

If an asymptotic expansion has first nonzero coefficient $a_k$ , its leading term is $a_k\phi_k$ . For scalar coefficients,

f(\varepsilon)=a_k\phi_k(\varepsilon)+o(|\phi_k(\varepsilon)|), \qquad f\sim a_k\phi_k.

The coefficient and comparison function together give the dominant approximation.

Example

If $f(\varepsilon)=3\varepsilon^2-\varepsilon^3+O(\varepsilon^4)$ , the leading term is $3\varepsilon^2$ . An expansion with all coefficients zero has no first nonzero term; a nonzero flat function can have such an expansion.