integral de residu imperial

lim[ a --> 0 ][ int[ 0-->2pi ][ z = ae^{ix} ][ f(z)/z^{(1/m)} ] d_{x}^{(1/m)}[z] d[x] ] = ...

... (2pi)·i^{(1/m)}·f(0)

d_{x}^{(1/m)}[x] = x^{(1/m)+(-1)·(1/m)} = x^{0} = 1

lim[ a --> 0 ][ int[ 0-->2pi ][ z = ae^{ix} ][ f(z)/z ] d_{x}[z] d[x] ] = 2pi·i·f(0) 

lim[ a --> 0 ][ int[ 0-->2pi ][ z = g^{o(-1)}(ae^{ix}) ][ f(z)/( g(z) )^{(1/m)} ] d_{x}^{(1/m)}[z] d[x] ] = ...

...  (2pi)·i^{(1/m)}·f( g^{o(-1)}(0) )·d_{0}^{(1/m)}[g^{o(-1)}(0)]

lim[ a --> 0 ][ int[ 0-->2pi ][ z = g^{o(-1)}(ae^{ix}) ][ f(z)/g(z) ] d_{x}[z] d[x] ] = ...

...  2pi·i·f( g^{o(-1)}(0) )·d_{0}[g^{o(-1)}(0)]

lim[ a --> 0 ][ int[ 0-->2pi ][ z = e^{ae^{ix}} ][ f(z)/ln(z) ] d_{x}[z] d[x] ] = ...

...  2pi·i·f(1)

lim[ a --> 0 ][ int[ 0-->2pi ][ z = ln(ae^{ix}) ][ f(z)/e^{z} ] d_{x}[z] d[x] ] = ...

...  2pi·i·f( ln(0) )·oo

lim[ a --> 0 ][ int[ 0-->2pi ][ z = e^{ae^{ix}} ][ f(z)/( ln(z) )^{(1/m)} ] d_{x}^{(1/m)}[z] d[x] ] = ...

...  2pi·i^{(1/m)}·f(1)

lim[ a --> 0 ][ int[ 0-->2pi ][ z = ln(ae^{ix}) ][ f(z)/( e^{z} )^{1/m} ] d_{x}^{(1/m)}[z] d[x] ] = ...

...  2pi·i^{(1/m)}·f( ln(0) )·oo^{(1/m)}

lim[ a --> 0 ][ int[ 0-->2pi ][ z = (ae^{ix})^{(1/n)} ][ f(z)/z^{n} ] d_{x}[z] d[x] ] = ...

...  2pi·i·f( 0^{(1/n)} )·(1/n)·0^{(1/n)}·oo

lim[ a --> 0 ][ int[ 0-->2pi ][ z = (ae^{ix})^{n} ][ f(z)/z^{(1/n)} ] d_{x}[z] d[x] ] = ...

...  2pi·i·f( 0^{n} )·n·0^{n}·oo

lim[ a --> 0 ][ int[ 0-->2pi ][ z = (ae^{ix})^{(1/n)} ][ f(z)/( z^{n} )^{(1/m)} ] d_{x}^{(1/m)}[z] d[x] ] = ...

...  2pi·i^{(1/m)}·f( 0^{(1/n)} )·(1/n^{(1/m)})·0^{(1/m)·(1/n)}·oo^{(1/m)}

lim[ a --> 0 ][ int[ 0-->2pi ][ z = (ae^{ix})^{n} ][ f(z)/( z^{(1/n)} )^{(1/m)} ] d_{x}^{(1/m)}[z] d[x] ] = ...

...  2pi·i^{(1/m)}·f( 0^{n} )·n^{(1/m)}·0^{(1/m)·n}·oo^{(1/m)}