CV9

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List all the cube roots of z={\sqrt  {2}}+i{\sqrt  {2}}\,.

Let z=x+iy\, and x=r\cos \theta ,y=r\sin \theta ,r={\sqrt  {x^{2}+y^{2}}}\,

r=|z|=|{\sqrt  {2}}+i{\sqrt  {2}}|=2\,

2\cos \theta ={\sqrt  {2}}\implies \theta ={\frac  {\pi }{4}}\,

2\sin \theta ={\sqrt  {2}}\implies \theta ={\frac  {\pi }{4}}\,

So z=2e^{{i{\frac  {\pi }{4}}}}\,.

The {\mathrm  {m^{{th}}}}\, roots of a complex number written in polar form z=|z|e^{{i\theta }}\, are

z^{{{\frac  {1}{m}}}}=|z|^{{\frac  {1}{m}}}\exp \left({\frac  {i(\theta +2k\pi )}{m}}\right),k=0,1,2,...,m-1\,

So the cube roots of 2e^{{i{\frac  {\pi }{4}}}}\, are

2^{{1/3}}\exp \left({\frac  {i(\pi /4+2k\pi )}{3}}\right),k=0,1,2\,

=2^{{1/3}}e^{{i{\frac  {\pi }{12}}}},2^{{1/3}}e^{{i{\frac  {9\pi }{12}}}},2^{{1/3}}e^{{i{\frac  {17\pi }{12}}}}\,.

Complex Variables

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