Consider the functionh(z) =2iz+1 defined on the extended complex plane. The functionhcan beviewed as a composition...

Consider the functionh(z) =2iz+1 defined on the extended complex plane. The functionhcan beviewed as a composition of a linear mapping and the reciprocal mapping; explicitly,h(z) = (g◦f)(z) =g(f(z)),wherefis the reciprocal mappingf(z) =1/zandgis the linear mappingg(z) =2i z+1

.(a)Describe in words the action of the mappingh.

(b)Determine the image of the line Rez=4 under the mappingw=h(z). Sketch the line in thezplane and its image in thewplane.

(c)Determine the image of the circle|z+i|=1/2 under the mappingw=h(z). Sketch the circle inthezplane and its image in thewplane.

[You can use MAPLEor MATLABto generate plots if you so choose.]

Expert Solution

Consider the functionh(z) =2iz+1 defined on the extended complex plane. The functionhcan beviewed as a composition of a linear mapping and the reciprocal mapping; explicitly,h(z) = (g◦f)(z) =g(f(z)),wherefis the reciprocal mappingf(z) =1/zandgis the linear mappingg(z) =2i z+1

Colby Messinger answered 1 month ago

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