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IntegrationTools

 ExpandMultiple
 expand a multiple integral into a nested integral

 Calling Sequence ExpandMultiple(v) ExpandMultiple(v, stripoptions)

Parameters

 v - multiple integral

Description

 • The function ExpandMultiple expands a multiple integral given as a single Int function call into a nested series of Int function calls.
 • ExpandMultiple will also work similarly on unevaluated int function calls.
 • This command is primarily intended for programmers who need to manipulate integrals as data structures.  In most cases, multiple integrals should be computed from a single function call.  The output of ExpandMultiple can be transformed back into a single function call using IntegrationTools[CollapseNested].
 • If stripoptions is given, none of the options in the integral v will appear in the output
 • If v is a nested Int or int function call, only the outermost function call will be expanded.

Examples

 > $\mathrm{with}\left(\mathrm{IntegrationTools}\right):$
 > $\mathrm{lprint}\left(\mathrm{Int}\left(f\left(x,y\right),\left[x,y\right]\right)\right)$
 Int(f(x,y),[x, y])
 > $\mathrm{ExpandMultiple}\left(\mathrm{Int}\left(f\left(x,y\right),\left[x,y\right]\right)\right)$
 ${∫}{∫}{f}{}\left({x}{,}{y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}$ (1)
 > $\mathrm{lprint}\left(\right)$
 Int(Int(f(x,y),x),y)
 > $\mathrm{ExpandMultiple}\left(\mathrm{Int}\left(f\left(x,y\right),x,y\right)\right)$
 ${∫}{∫}{f}{}\left({x}{,}{y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}$ (2)
 > $\mathrm{lprint}\left(\right)$
 Int(Int(f(x,y),x),y)
 > $\mathrm{ExpandMultiple}\left({{∫}}_{c}^{d}{{∫}}_{a}^{b}f\left(x,y\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x{ⅆ}y\right)$
 ${{∫}}_{{c}}^{{d}}{{∫}}_{{a}}^{{b}}{f}{}\left({x}{,}{y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}$ (3)
 > $\mathrm{lprint}\left(\right)$
 Int(Int(f(x,y),x = a .. b),y = c .. d)
 > $\mathrm{CollapseNested}\left(\right)$
 ${{∫}}_{{c}}^{{d}}{{∫}}_{{a}}^{{b}}{f}{}\left({x}{,}{y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}$ (4)
 > $\mathrm{lprint}\left(\right)$
 Int(f(x,y),[x = a .. b, y = c .. d])
 > $\mathrm{ExpandMultiple}\left('\mathrm{int}'\left(f\left(x,y\right),\left[x=a..b,y=c..d\right]\right)\right)$
 ${{∫}}_{{c}}^{{d}}{{∫}}_{{a}}^{{b}}{f}{}\left({x}{,}{y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}$ (5)
 > $\mathrm{lprint}\left(\right)$
 int(int(f(x,y),x = a .. b),y = c .. d)
 > $\mathrm{ExpandMultiple}\left(\mathrm{Int}\left(f\left(x,y\right),\left[x=a..b,y=c..d\right],\mathrm{CauchyPrincipalValue}\right)\right)$
 ${\mathrm{Int}}{}\left({\mathrm{Int}}{}\left({f}{}\left({x}{,}{y}\right){,}{x}{=}{a}{..}{b}{,}{\mathrm{CauchyPrincipalValue}}\right){,}{y}{=}{c}{..}{d}{,}{\mathrm{CauchyPrincipalValue}}\right)$ (6)
 > $\mathrm{lprint}\left(\right)$
 Int(Int(f(x,y),x = a .. b,CauchyPrincipalValue),y = c .. d,CauchyPrincipalValue)
 > $\mathrm{ExpandMultiple}\left(\mathrm{Int}\left(f\left(x,y\right),\left[x=a..b,y=c..d\right],\mathrm{CauchyPrincipalValue}\right),\mathrm{stripoptions}\right)$
 ${{∫}}_{{c}}^{{d}}{{∫}}_{{a}}^{{b}}{f}{}\left({x}{,}{y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}$ (7)
 > $\mathrm{lprint}\left(\right)$
 Int(Int(f(x,y),x = a .. b),y = c .. d)