# q calculus

Table of Contents

Quantum calculus, typically referred to as calculus with out limits, is equal to conventional infinitesimal calculus with out the notion of limits. It defines “q-calculus” and “h-calculus”, the place h ostensibly stands for Planck’s fixed whereas q stands for quantum. The 2 parameters are associated by the formulation

the place

=

h

2
π

{displaystyle scriptstyle hbar ={frac {h}{2pi }},}

is the decreased Planck fixed.

## Differentiation

Within the q-calculus and h-calculus, differentials of features are outlined as

and

respectively. Derivatives of features are then outlined as fractions by the q-derivative

and by

Within the restrict, as h goes to 0, or equivalently as q goes to 1, these expressions tackle the type of the spinoff of classical calculus.

## Integration

q-integral

A operate F(x) is a q-antiderivative of f(x) if DqF(x) = f(x). The q-antiderivative (or q-integral) is denoted by

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f
(
x
)

d

q

x

{displaystyle int f(x),d_{q}x}

and an expression for F(x) could be discovered from the formulation

f
(
x
)

d

q

x
=
(
1

q
)

j
=

x

q

j

f
(
x

q

j

)

{displaystyle int f(x),d_{q}x=(1-q)sum _{j=0}^{infty }xq^{j}f(xq^{j})}

which is named the Jackson integral of f(x). For 0 < q < 1, the sequence converges to a operate F(x) on an interval (0,A] if |f(x)xα| is bounded on the interval (0,A] for some 0 ≤ α < 1. The q-integral is a Riemann–Stieltjes integral with respect to a step operate having infinitely many factors of improve on the factors qj, with the bounce on the level qj being qj. If we name this step operate gq(t) then dgq(t) = dqt.[1] h-integral A operate F(x) is an h-antiderivative of f(x) if DhF(x) = f(x). The h-antiderivative (or h-integral) is denoted by ∫ f ( x ) d h x {displaystyle int f(x),d_{h}x} . If a and b differ by an integer a number of of h then the particular integral ∫ a b f ( x ) d h x {displaystyle int _{a}^{b}f(x),d_{h}x} is given by a Riemann sum of f(x) on the interval [a,b] partitioned into subintervals of width h.

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## Instance

The spinoff of the operate

x

n

{displaystyle x^{n}}

(for some constructive integer

n

{displaystyle n}

) within the classical calculus is

n

x

n

1

{displaystyle nx^{n-1}}

. The corresponding expressions in q-calculus and h-calculus are

with the q-bracket

and

respectively. The expression

[
n

]

q

x

n

1

{displaystyle [n]_{q}x^{n-1}}

is then the q-calculus analogue of the easy energy rule for
constructive integral powers. On this sense, the operate

x

n

{displaystyle x^{n}}

continues to be good within the q-calculus, however somewhat
ugly within the h-calculus – the h-calculus analog of

x

n

{displaystyle x^{n}}

is as an alternative the falling factorial,

(
x

)

n

:=
x
(
x

1
)

(
x

n
+
1
)
.

{displaystyle (x)_{n}:=x(x-1)cdots (x-n+1).}

One could proceed additional and develop, for instance, equal notions of Taylor enlargement, et cetera, and even arrive at q-calculus analogues for the entire standard features one would need to have, akin to an analogue for the sine operate whose q-derivative is the suitable analogue for the cosine.

## Historical past – “q calculus”

The h-calculus is simply the calculus of finite variations, which had been studied by George Boole and others, and has confirmed helpful in a variety of fields, amongst them combinatorics and fluid mechanics. The q-calculus, whereas courting in a way again to Leonhard Euler and Carl Gustav Jacobi, is barely lately starting to see extra usefulness in quantum mechanics, having an intimate reference to commutativity relations and Lie algebra.

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“q calculus”