1-z-a-bi-2-z-re-i-express-the-values-of-a-real-z-z-real-part-b-imag-z-z-imaginary-part-c-abs-z-z-absolute-value-d-arg-z-z-argument-angle-

Question Number 67992 by MJS last updated on 03/Sep/19

(1) z = a + b i

(2) z = r e^{i θ}

express the values of

(a) real (z^{z}) [real part]

(b) imag (z^{z}) [imaginary part]

(c) abs (z^{z}) [absolute value]

(d) \arg (z^{z}) [argument = angle]

Answered by mr W last updated on 03/Sep/19

l e t A = z^{z}

\ln A = z \ln z = (a + b i) (\ln r + i θ)

= (a \ln r - b θ) + (a θ + b \ln r) i

A = e^{a \ln r - b θ} e^{(a θ + b \ln r) i}

= e^{a \ln r - b θ} {\cos (a θ + b \ln r) + i \sin (a θ \times b \ln r)}

real (z^{z}) = e^{a \ln r - b θ} \cos (a θ + b \ln r)

i m a g (z^{z}) = e^{a \ln r - b θ} \sin (a θ + b \ln r)

a b s (z^{z}) = e^{a \ln r - b θ}

a r g (z^{z}) = a θ + b \ln r

Commented by MJS last updated on 03/Sep/19

thank you

funny, I did it exactly the same way, had been

unsure if I^{'} m right \dots

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