Lecture 7: Polar representation of complex numbers

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1 Lecture 7: Polar represetatio of comple umbers See FLAP Module M3.1 Sectio.7 ad M3. Sectios 1 ad. 7.1 The Argad diagram I two dimesioal Cartesia coordiates (,), we are used to plottig the fuctio ( ) with o the vertical ais ad o the horizotal ais. I a Argad diagram, the comple umber z i is plotted as a sigle poit with coordiates (,). The horizotal ais is called the real ais (-ais) ad the vertical ais is called the imagiar ais (-ais). As i usual Cartesia coordiates, the distace from the origi to a poit (, ) is equal to. This is equal to the modulus z of the comple umber z i. The Argad diagram ma also be called the comple plae. It stresses that comple umbers are a geeralisatio of real umbers, that lie o the horizotal ais ol.

2 The epressio z i is kow as the Cartesia form or the rectagular form of the comple umber z. Usig the Argad diagram, we ca see that the additio of comple umbers behaves like the additio of vectors. If we epress z i,, the the additio of two comple umbers ma be defied b as a ordered pair ( ) (, ) ( a, b) ( a, b) i the same wa as the additio of two vectors.

3 7. Polar coordiates A positio vector of a poit i two dimesios ma be epressed i terms of Cartesia coordiates (,) ad plotted with o the vertical ais ad o the horizotal ais. It is also possible to epress the two dimesioal positio vector i terms of polar coordiates (r,θ) where r is the magitude of the vector (distace from origi to the poit) ad θ is the agle betwee the positio vector ad the positive -ais. The Cartesia ad polar coordiates are related b: r cosθ, r siθ r, taθ

4 I the same wa, the comple umber z i ma be epressed i polar coordiates ( r,θ ) i its polar form: where ( cosθ isiθ ) z i r r, taθ e. g. z 1 3i 1, 3 r taθ 3 4 θ π / 3 ( 60 )

5 7.3 The modulus of a comple umber r the magitude r of the distace from the origi to the poit represeted b z is equal to the modulus I polar coordiates (,θ ) of the comple umber z : r z 7.4 The argumet of a comple umber I polar coordiates ( r,θ ) the agle θ is kow as the argumet of the comple umber z, deoted arg( z) θ. cosθ, siθ There is a complicatio because a sigle poit o the Argad diagram does ot correspod to a sigle comple umber. The reaso is that we ca add π to the value of the argumet θ i order to produce a differet comple umber, but whe plotted o the Argad diagram, the two umbers are plotted i the same place.

6 Pricipal value: If we wat to uiquel defie the value of the argumet θ we ca impose the coditio π < θ π so that θ is kow as the pricipal value of the argumet. For the comple umber z i, the argumet θ is give b the solutio of the equatios: or cosθ, siθ taθ If the secod epressio ta θ / is used to determie θ, it is wise to plot z i o a Argad diagram to check that the aswer is correct (see page 7, FLAP Module M3.).

7 7.5 Euler s formula A useful epressio is Euler s formula which epresses a epoetial with imagiar argumet i terms of a sum of real ad imagiar parts: θ e i cosθ i si θ We ca see where this comes from usig the series represetatios of the epoetial, sie ad cosie fuctios (PHYS14): 3 4 ep( ) 1... for all! 1!! 3! 4! 0 si( ) 0 ( 1) ( ) 1! 1 3 3! 5 5! 7 7!... for all ( 1) ( ) 4 6 cos( ) 1... for all 0!! 4! 6! If we eted the epoetial series epressio for imagiar argumet z iθ where θ is real, the 3 4 ( iθ ) iθ ( iθ ) ( iθ ) ( iθ ) ep( iθ ) ! 1!! 3! 4! 3 4 θ θ θ θ 1 i i K 1!! 3! 4! 4 3 θ θ θ θ 1 K i! 4! K 1! 3! cosθ i siθ

8 Epoetial form: Usig Euler s formula, it is possible to compactl write a comple umber i terms of a epoetial fuctio: ( cos θ i θ ) re iθ z i r si O a Argad diagram, comple umbers with the same modulus r z but differet argumets θ make up poits o a circle cetred o the origi with radius r r 1, the circle is of radius equal to oe. Special poits of iterest (for r 1) are: z. For modulus θ 0 θ θ π θ π 3π θ π z e i0 z e z e z e i z e i π iπ 3π iπ 1 cos i si cosπ i si π 1 cos π 3π i si π 3π i i cos π i si π 1 real imagiar real imagiar real

1. Complex numbers. Chapter 13: Complex Numbers. Modulus of a complex number. Complex conjugate. Complex numbers are of the form

1. Complex numbers. Chapter 13: Complex Numbers. Modulus of a complex number. Complex conjugate. Complex numbers are of the form Comple umbers ad comple plae Comple cojugate Modulus of a comple umber Comple umbers Comple umbers are of the form Sectios 3 & 32 z = + i,, R, i 2 = I the above defiitio, is the real part of z ad is the