FeynCalc manual (development version)

Convolute

Convolute[f, g, x] convolutes f(x) and g(x), i.e., \int _0^1 dx_1 \int _0^1 dx_2 \delta \left(x - x_1 x_2\right) f (x_1) g(x_2).

Convolute[f, g] is equivalent to Convolute[f, g, x].

Convolute[exp, {x1, x2}] assumes that exp is polynomial in x1 and x2. Convolute uses table-look-up and does not do any integral calculations, only linear algebra.

See also

Overview, PlusDistribution, ConvoluteTable.

Examples

Convolute[1, 1] /. FCGV[z_] :> ToExpression[z]

-\log (x)

Convolute[x, x] /. FCGV[z_] :> ToExpression[z]

-x^2 \log (x)

Convolute[1, x] /. FCGV[z_] :> ToExpression[z]

-x \log (x)

Convolute[1, 1/(1 - x)] /. FCGV[z_] :> ToExpression[z]

\frac{\log (x)}{x-1}

Convolute[1, PlusDistribution[1/(1 - x)]] /. FCGV[z_] :> ToExpression[z]

\frac{\log (x)}{x-1}

Convolute[1/(1 - x), x] /. FCGV[z_] :> ToExpression[z]

\frac{x \log (x)}{x-1}

Convolute[1/(1 - x), 1/(1 - x)] /. FCGV[z_] :> ToExpression[z]

-\frac{\log (x)}{(x-1)^2}

Convolute[1, Log[1 - x]] /. FCGV[z_] :> ToExpression[z]

-\log (1-x) \log (x)

Convolute[1, x Log[1 - x]] /. FCGV[z_] :> ToExpression[z]

-x \log (1-x) \log (x)

Convolute[1/(1 - x), Log[1 - x]] /. FCGV[z_] :> ToExpression[z]

\frac{\log (1-x) \log (x)}{x-1}

Convolute[1/(1 - x), x Log[1 - x]] /. FCGV[z_] :> ToExpression[z]

\frac{x \log (1-x) \log (x)}{x-1}

Convolute[Log[1 - x]/(1 - x), x] /. FCGV[z_] :> ToExpression[z]

\frac{x \log (1-x) \log (x)}{x-1}

Convolute[1, x Log[x]] /. FCGV[z_] :> ToExpression[z]

-x \log ^2(x)

Convolute[Log[1 - x], x] /. FCGV[z_] :> ToExpression[z]

-x \log (1-x) \log (x)

Convolute[1/(1 - x), Log[x]/(1 - x)] /. FCGV[z_] :> ToExpression[z]

-\frac{\log ^2(x)}{(x-1)^2}

Convolute[1, Log[x]] /. FCGV[z_] :> ToExpression[z]

-\log ^2(x)

Convolute[x, x Log[x]] /. FCGV[z_] :> ToExpression[z]

-x^2 \log ^2(x)

Convolute[1/(1 - x), Log[x]] /. FCGV[z_] :> ToExpression[z]

\frac{\log ^2(x)}{x-1}

Convolute[1, Log[x]/(1 - x)] /. FCGV[z_] :> ToExpression[z]

\frac{\log ^2(x)}{x-1}

Convolute[1/(1 - x), x Log[x]] /. FCGV[z_] :> ToExpression[z]

\frac{x \log ^2(x)}{x-1}

Convolute[Log[x]/(1 - x), x] /. FCGV[z_] :> ToExpression[z]

\frac{x \log ^2(x)}{x-1}

Convolute[1, x Log[x]] /. FCGV[z_] :> ToExpression[z]

-x \log ^2(x)

Convolute[Log[x], x] /. FCGV[z_] :> ToExpression[z]

-x \log ^2(x)

Convolute[1/(1 - x), Log[1 - x]/(1 - x)] /. FCGV[z_] :> ToExpression[z]

-\frac{\log (1-x) \log (x)}{(x-1)^2}

Convolute[Log[1 - x]/(1 - x), Log[1 - x]] /. FCGV[z_] :> ToExpression[z]

\frac{\log ^2(1-x) \log (x)}{x-1}