Fungsi Kompleks f(z) = exp(z)

Fungsi kompleks f(z):=e^z memetakan z=x+iy ke w=u+iv dengan u=e^x\cos y dan v=e^x\sin y, karena

e^{x+iy}=e^xe^{iy}=e^x(\cos y+i\sin y).

Perhatikan jika x=k (k konstanta), maka

|e^z|=e^x=e^k (karena |\cos y + i\sin y|=1).

Jadi f memetakan garis-garis vertikal x=k ke lingkaran-lingkaran |w|=e^k.

Khususnya, f memetakan garis x=0 (sumbu imajiner) ke lingkaran satuan |w|=1.

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Bandung, 30-06-2020

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