From Example Problems
Jump to: navigation, search

eve navarro videos map old movie video mobile video card gpu comparison alpha dog movie watch azurus torrent downloader anjali free movies mandys movie animal care training videos inspirational scripture porn cartoon movie anara clip gupta in india video alien scary video nude movie clip disney little mermaid movie index new movie news poseidon adventure movie smart movie converter cracked movietheaters nyc evolution video atlanta http mike jones back then music video ice age movie font schism music video download joe dirt movie clips marvin gaye movie ab normal beauty movie review little shop of horror movie indian porn star movie mustang street racing videos ray lewis pregame dance video integrated video los video mas sorprendentes del mundo radio controlled car video september 11th news video lego movie pickett australian movie releases attacking mule tourist video movie theater guymon marbles video game movie dallas activation check compose ctrl mail search movie favourites kittie videos on launch screaming woman video intermission movie short eminems music video just lose it pirate radio xm superduke movie page movieland virus phpedit license file private porn movie .com question for stand and deliver movie sports illustrated video clip off licenses in uk movie countdown leader radio movie movie trailer music download nicole ritchie flash video play videos online free ass fucking movie pocket pc free video alkaline trio video streaming music videos online http maketorrent 2.0 straylight run video codes inspirational poem updated park 12 cobb movies la rush no cd peter gabriel sledgehammer video free lesbian sample sex video inspiring flash movies rent the movie premier realviz stitcher 5 crack spongebob hemppants video jake gyllenhaal movie movie rating logo movies summerville sc naked cartoon movie away rasmus sail video page 3 movie download into the moat video shake your bon bon video pirate ship kid american journal of respiratory care medicine omarion o videos lesbians making out video show me the money movie only the strong survive movie sitemap links spectrum movie theater albany movie vallavan andrew lloyd weber phantom of the opera movie online video how to educational software license killing us softly iii video rabbiting videos mister mom movie khalsa junction videos http webmap jin video battles jumpers hole road movie theater engine online search video medical teaching videos renting adult movie pasroolocale Find the residues of f(z)\, at all its isolated singular points and at infinity (if infinity is not a limit point of singular points), where f(z)\, is given by

f(z)={\frac  {1}{z(1-z^{2})}}\,

This has poles at z=0,\pm 1\, of multiplicity k=1\,.

Use the formula {\mathrm  {Res}}_{{z=z_{0}}}f(z)={\frac  {1}{(k-1)!}}\lim _{{z\to z_{0}}}{\frac  {d^{{k-1}}}{{dz}^{{k-1}}}}\left[(z-z_{0})^{k}f(z)\right]\,

{\mathrm  {Res}}_{{z=-1}}=\lim _{{z\to -1}}(z+1)f(z)=\lim _{{z\to -1}}{\frac  {1}{z(1-z)}}={\frac  {-1}{2}}\,

{\mathrm  {Res}}_{{z=0}}=\lim _{{z\to 0}}zf(z)=\lim _{{z\to 0}}{\frac  {1}{(1-z^{2})}}=1\,

{\mathrm  {Res}}_{{z=1}}=\lim _{{z\to 1}}(z-1)f(z)=\lim _{{z\to 1}}{\frac  {-1}{z(1+z)}}={\frac  {-1}{2}}\,

If f(z)\, is analytic except at isolated singular points, then the sum of all the residues of f(z)\, equals 0.

So {\mathrm  {Res}}_{{z=0}}+{\mathrm  {Res}}_{{z=-1}}+{\mathrm  {Res}}_{{z=1}}+{\mathrm  {Res}}_{{z=\infty }}=1-{\frac  {1}{2}}-{\frac  {1}{2}}+{\mathrm  {Res}}_{{z=\infty }}=0\, and {\mathrm  {Res}}_{{z=\infty }}=0\,

Main Page : Complex Variables : Residues