= Solution
For an infinite <cardinal number> $\kappa$, the <gimel function> is
$$
\boxed{\gimel(\kappa)=\kappa^{\operatorname{cf}(\kappa)}}.
$$
The <singular cardinals hypothesis> asserts that for every infinite <singular cardinal> $\kappa$,
$$
\boxed{2^{\operatorname{cf}(\kappa)}<\kappa
\ \Longrightarrow\ \gimel(\kappa)=\kappa^+.}
$$
Here $\kappa^+$ is the <successor cardinal>. Equivalently, for every infinite <singular cardinal>,
$$
\kappa^{\operatorname{cf}(\kappa)}
=\max\{\kappa^+,2^{\operatorname{cf}(\kappa)}\}.
$$
To see why the second formulation adds nothing in the other case, put $\theta=\operatorname{cf}(\kappa)$. If $\kappa\leq2^\theta$, then $2^\theta\leq\kappa^\theta\leq(2^\theta)^\theta=2^\theta$ by <infinite cardinal arithmetic>. Equality $\kappa=2^\theta$ is impossible here: the <König theorem for cardinal numbers> gives $\operatorname{cf}(2^\theta)>\theta$, whereas $\operatorname{cf}(\kappa)=\theta$. Thus in this case $2^\theta\geq\kappa^+$, as required by the maximum formula. For a <strong limit cardinal> that is singular, the hypothesis also yields $2^\kappa=\kappa^+$: restrictions of a subset of $\kappa$ to a cofinal sequence of smaller cardinals give $2^\kappa\leq\kappa^\theta$, while the reverse inequality is immediate. The quantified implication above is the precise general statement.
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