MATH2069Chapter01.pdf - Chapter 1 The Complex Plane 1.1...

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Chapter 1The Complex Plane1.1 Revision of First Year1.2 Topology of the Complex Plane1.3 Mapping by Complex FunctionsMATH2069 Complex Notes,2016– p. 11.1 Revision of First YearWe use the symbolias the square root of1, so the othersquare root isi.(You can usejif you want.)Acomplex number, usually denotedz, is an expression ofthe formx+iyfor real numbersxandy.We callxthereal part, denoted(z)orRe(z), andytheimaginary part, denoted(z)orIm(z).We will from here implicitly use this notation, so for anycomplex numberz, the symbolxwill, in the absence of anyother definition, mean the real part ofzand similarly fory.We denote the set of all complex numbers byC.MATH2069 Complex Notes,2016– p. 2
MATH2069 Complex Notes,2016– p. 4
MATH2069 Complex Notes,2016– p. 6
Before some examples, here are some easy results linking ourfunctions:a)z+z=2(z)andzz=2i(z)b)|z|=|z|andzz=|z|2c)z+w=z+wandzw=z·wd)1z=z|z|2e) Ifz=r ethenz=r e-f) Ifz=r eandw=s ethenzw=rs ei(θ+φ).They’re all too easy for me to need to prove: make sure youcan prove them all.MATH2069 Complex Notes,2016– p. 7Example 2Putiand22i

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