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https://texercises.com/exercise/complex-integrals-3/
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Exercise:
Compute the following egral: _gamma fracz+z^z+ddz where gamma is the circle centered at with radius counterclockwise oriented.

Solution:
The egrand function fz has a simple pole at z- and a pole of order at z. For the residue at z- we get: textRes_-f lim_zrightarrow -fracz+z^ frac-+- - For the one at z we can do the calculation: textRes_f frac!lim_zrightarrow leftfracdddd zright^ fracz+z+ frac!lim_zrightarrow leftfracdddd zright^ frac-z+^ frac!lim_zrightarrow leftfracdddd zright^ fracz+^ frac!lim_zrightarrow leftfracdddd zright frac- z+^ frac!lim_zrightarrow frac z+^ lim_zrightarrow fracz+^ Else we can use the taylor series as a shortcut: fracz^fracz+z+ fracz+z^-z+z^-z^+... fracz^z-z^+z^-z^+-z+z^-z^+z^+... fracz^...+z^-+... ...+fracz+... &Rightarrow textRes_f Either way we get a total result of _gamma fracz+z^z+ddz pi i -
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Exercise:
Compute the following egral: _gamma fracz+z^z+ddz where gamma is the circle centered at with radius counterclockwise oriented.

Solution:
The egrand function fz has a simple pole at z- and a pole of order at z. For the residue at z- we get: textRes_-f lim_zrightarrow -fracz+z^ frac-+- - For the one at z we can do the calculation: textRes_f frac!lim_zrightarrow leftfracdddd zright^ fracz+z+ frac!lim_zrightarrow leftfracdddd zright^ frac-z+^ frac!lim_zrightarrow leftfracdddd zright^ fracz+^ frac!lim_zrightarrow leftfracdddd zright frac- z+^ frac!lim_zrightarrow frac z+^ lim_zrightarrow fracz+^ Else we can use the taylor series as a shortcut: fracz^fracz+z+ fracz+z^-z+z^-z^+... fracz^z-z^+z^-z^+-z+z^-z^+z^+... fracz^...+z^-+... ...+fracz+... &Rightarrow textRes_f Either way we get a total result of _gamma fracz+z^z+ddz pi i -
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ca, complex, integral, poles, residue, substitution
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ENG (English)
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