= Leading Taylor term
{title2=$a_m(z-c)^m$}
Suppose an <analytic function> near $c$ is not identically zero there. Its <Taylor series> has a first nonzero <coefficient> $a_m$, and
$$
f(z)=a_m(z-c)^m+O((z-c)^{m+1}).
$$
The term $a_m(z-c)^m$ is its <leading Taylor term>. Dividing by $(z-c)^m$ gives a <limit> of $a_m\ne0$ 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 $f(c)\ne0$.
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