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Question Number 148639 by tabata last updated on 29/Jul/21

find singular point of this following and whats the type of singular point ? (1)f(z)=(1/(lnz)) (2)f(z)=((1−cos(z+i))/(z(z^2 +1)^2 )) (3)f(z)=((sinz)/(z^2 +z)) (4)f(z)=((sin2z)/z^2 )

$${find}\:{singular}\:{point}\:{of}\:{this}\:{following}\:{and} \\ $$$${whats}\:{the}\:{type}\:{of}\:{singular}\:{point}\:? \\ $$$$ \\ $$$$\left(\mathrm{1}\right){f}\left({z}\right)=\frac{\mathrm{1}}{{lnz}} \\ $$$$ \\ $$$$\left(\mathrm{2}\right){f}\left({z}\right)=\frac{\mathrm{1}−{cos}\left({z}+{i}\right)}{{z}\left({z}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$ \\ $$$$\left(\mathrm{3}\right){f}\left({z}\right)=\frac{{sinz}}{{z}^{\mathrm{2}} +{z}} \\ $$$$ \\ $$$$\left(\mathrm{4}\right){f}\left({z}\right)=\frac{{sin}\mathrm{2}{z}}{{z}^{\mathrm{2}} } \\ $$

Answered by mathmax by abdo last updated on 30/Jul/21

1) f(z)=(1/(lnz)) o is singular point 2)f(z)=((1−cos(z+i))/(z(z−i)^2 (z+i)^2 )) les points singuliers sont les poles o ,+^− i 3)f(z)=((sinz)/(z(z+1))) ⇒les point singuliers sont o et −1 4)f(z)=((sin(2z))/z^2 ) ⇒o est point singulier (pole dordre 2)

$$\left.\mathrm{1}\right)\:\mathrm{f}\left(\mathrm{z}\right)=\frac{\mathrm{1}}{\mathrm{lnz}}\:\:\:\mathrm{o}\:\mathrm{is}\:\mathrm{singular}\:\mathrm{point} \\ $$$$\left.\mathrm{2}\right)\mathrm{f}\left(\mathrm{z}\right)=\frac{\mathrm{1}−\mathrm{cos}\left(\mathrm{z}+\mathrm{i}\right)}{\mathrm{z}\left(\mathrm{z}−\mathrm{i}\right)^{\mathrm{2}} \left(\mathrm{z}+\mathrm{i}\right)^{\mathrm{2}} }\:\:\:\mathrm{les}\:\mathrm{points}\:\mathrm{singuliers}\:\mathrm{sont}\:\mathrm{les}\:\mathrm{poles} \\ $$$$\mathrm{o}\:,\overset{−} {+}\mathrm{i} \\ $$$$\left.\mathrm{3}\right)\mathrm{f}\left(\mathrm{z}\right)=\frac{\mathrm{sinz}}{\mathrm{z}\left(\mathrm{z}+\mathrm{1}\right)}\:\Rightarrow\mathrm{les}\:\mathrm{point}\:\mathrm{singuliers}\:\mathrm{sont}\:\mathrm{o}\:\mathrm{et}\:−\mathrm{1} \\ $$$$\left.\mathrm{4}\right)\mathrm{f}\left(\mathrm{z}\right)=\frac{\mathrm{sin}\left(\mathrm{2z}\right)}{\mathrm{z}^{\mathrm{2}} }\:\Rightarrow\mathrm{o}\:\mathrm{est}\:\mathrm{point}\:\mathrm{singulier}\:\left(\mathrm{pole}\:\mathrm{dordre}\:\mathrm{2}\right) \\ $$

Commented by Sozan last updated on 30/Jul/21

thank you sir can you give me the series ajust number (1) please

$${thank}\:{you}\:{sir}\:{can}\:{you}\:{give}\:{me}\:{the}\:{series}\: \\ $$$${ajust}\:{number}\:\left(\mathrm{1}\right)\:{please} \\ $$

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