“A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs; it is because they are made with ideas.”
― G.H. Hardy, A Mathematician's Apology

Let's define something...
Difference transform
\[\boxed{\boxed{\mathcal{D}\{f\}(a,b):=\int_a^b\left(f(\lfloor x\rfloor)-f(x)\right)\ \mathrm{d}x}}\]

Here an operator \(\mathcal{D}\) is acting on a function \(f\), describing the difference between an indiscrete sum and a discrete sum on a certain interval \(a\) to \(b\). This operator may play a pivotal role in asymptotic expansions, analytic continuation, etc.

#DifferenceTransform #DifferenceTransformOperator #TranformOperator #DifferenceTransformation #AsymptoticExpansion #AnalyticContinuation #Mathematics #GHHardy #Mathematicians #Definition #Prologue #Maths #Poet #Painter #Patterns #Beauty #BeautyOfPatterns