 Leading Taylor term

Suppose an analytic function near $c$ is not identically zero there. Its Taylor series has a first nonzero coefficient $a_m$, andThe term $a_m(z-c{)}^m$ is its leading Taylor term. Dividing by $(z-c{)}^m$ gives a limit of $a_m\ne 0$ as $z\to c$, so this term determines the leading local magnitude and angular behaviour. The remainder uses big O notation; it is smaller by one power near the centre.

For a real-valued analytic function of a real variable, an even integer $m$ gives the same nearby sign on both sides of $c$, while an odd integer $m$ gives opposite signs. For example, $f(x)=x^2(1+x)$ has leading Taylor term $x^2$ and is positive on both sides of zero sufficiently nearby; $g(x)=x^3(1+x)$ has leading Taylor term $x^3$ and changes sign there. The finite factor $1+x$ stays positive for $|x|<1$. The ordinary constant term is the leading one when $f(c)\ne 0$.