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M ( k ) = sinh ⁡ ⁡ ( k / 2 ) k / 2 , {\displaystyle M(k)={\frac {\sinh(k/2)}{k/2}},}

where sinh ⁡ ⁡ ( t ) {\displaystyle \sinh(t)} is the hyperbolic sine function.

The pulse function may also be expressed as a limit of a rational function:

Π Π ( t ) = lim n → → ∞ ∞ , n ∈ ∈ ( Z ) 1 ( 2 t ) 2 n + 1 . {\displaystyle \Pi (t)=\lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}.}

First, we consider the case where | t | < 1 2 . {\textstyle |t|<{\frac {1}{2}}.} Notice that the term ( 2 t ) 2 n {\textstyle (2t)^{2n}} is always positive for integer n . {\displaystyle n.} However, 2 t < 1 {\displaystyle 2t<1} and hence ( 2 t ) 2 n {\textstyle (2t)^{2n}} approaches zero for large n . {\displaystyle n.}

It follows that: lim n → → ∞ ∞ , n ∈ ∈ ( Z ) 1 ( 2 t ) 2 n + 1 = 1 0 + 1 = 1 , | t | < 1 2 . {\displaystyle \lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}={\frac {1}{0+1}}=1,|t|<{\tfrac {1}{2}}.}

Second, we consider the case where | t | > 1 2 . {\textstyle |t|>{\frac {1}{2}}.} Notice that the term ( 2 t ) 2 n {\textstyle (2t)^{2n}} is always positive for integer n . {\displaystyle n.} However, 2 t > 1 {\displaystyle 2t>1} and hence ( 2 t ) 2 n {\textstyle (2t)^{2n}} grows very large for large n . {\displaystyle n.}

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