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Question Number 124 by novrya last updated on 25/Jan/15
If C is circle ∣z∣=1. Then the value of ∫_C ((cos z)/(sin z)) dz =....
$${If}\:{C}\:{is}\:{circle}\:\mid{z}\mid=\mathrm{1}.\:{Then}\:{the}\:{value}\:{of} \\ $$$$\underset{{C}} {\int}\:\frac{{cos}\:{z}}{{sin}\:{z}}\:{dz}\:=…. \\ $$
Commented by 123456 last updated on 13/Dec/14
o teorema dos residuos seria bem util aqui. note que no interior do contorno C so a um polo simples(z=0) o resto fica como exercicio...
$$\mathrm{o}\:\mathrm{teorema}\:\mathrm{dos}\:\mathrm{residuos}\:\mathrm{seria}\:\mathrm{bem}\:\mathrm{util}\:\mathrm{aqui}. \\ $$$$\mathrm{note}\:\mathrm{que}\:\mathrm{no}\:\mathrm{interior}\:\mathrm{do}\:\mathrm{contorno}\:\mathrm{C}\:\mathrm{so}\:\mathrm{a}\:\mathrm{um}\:\mathrm{polo}\:\mathrm{simples}\left(\mathrm{z}=\mathrm{0}\right) \\ $$$$\mathrm{o}\:\mathrm{resto}\:\mathrm{fica}\:\mathrm{como}\:\mathrm{exercicio}… \\ $$
Answered by 123456 last updated on 14/Dec/14
we have that C have only one simple pole at z=0 then res(f,0)=lim_(x→0) ((xcos x)/(sin x))=1 then by residue theorem ∫_C ((cos z)/(sin z))dz=2πi res(f,0)=2πi
$$\mathrm{we}\:\mathrm{have}\:\mathrm{that}\:\mathrm{C}\:\mathrm{have}\:\mathrm{only}\:\mathrm{one}\:\mathrm{simple}\:\mathrm{pole}\:\mathrm{at}\:\mathrm{z}=\mathrm{0}\:\mathrm{then} \\ $$$$\mathrm{res}\left(\mathrm{f},\mathrm{0}\right)=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}\mathrm{cos}\:{x}}{\mathrm{sin}\:{x}}=\mathrm{1} \\ $$$$\mathrm{then}\:\mathrm{by}\:\mathrm{residue}\:\mathrm{theorem} \\ $$$$\int_{{C}} \frac{\mathrm{cos}\:{z}}{\mathrm{sin}\:{z}}{dz}=\mathrm{2}\pi{i}\:\mathrm{res}\left(\mathrm{f},\mathrm{0}\right)=\mathrm{2}\pi{i} \\ $$

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