If-f-z-z-sin-z-z-2-verify-if-f-z-satisfy-cauchy-rieman-condition-

Question Number 152771 by Tawa11 last updated on 01/Sep/21

If f (z) = z \sin (z) + ∣ z ∣^{2}, verify if f (z) satisfy cauchy rieman

condition

Commented by alisiao last updated on 01/Sep/21

f (z) = z s i n (z) + z \overline{z}

f (z) = (x + i y) s i n (x + i y) + x^{2} - y^{2}

f (z) = (x + i y) [s i n (x) c o s (i y) + c o s (x) s i n (i y)] + x^{2} - y^{2}

f (z) = (x + i y) [s i n (x) c o s h (y) + i c o s (x) s i n h (y)] + x^{2} - y^{2}

f (z) = (x s i n (x) c o s h (y) - y c o s (x) s i n h (y) + x^{2} - y^{2}] + i [y s i n (x) c o s h (y) + x c o s (x) s i n h (y)]

U (x, y) = x s i n (x) c o s h (y) - y c o s (x) s i n h (y) + x^{2} - y^{2}

V (x, y) = y s i n (x) c o s h (y) + x c o s (y) s i n h (y)

U_{x} = x c o s (x) c o s h (y) + s i n (x) c o s h (y) + y s i n (x) s i n h (y) + 2 x

U_{y} = y s i n (x) s i n h (y) + s i n (x) c o s h (y) - y c o s (x) c o s h (y) - c o s (x) s i n h (y) - 2 y

V_{x} = y c o s (x) c o s h (y) + c o s (y) s i n h (y)

V_{y} = y s i n (x) s i n h (y) + s i n (x) c o s h (y) + x c o s (y) c o s h (y) - x s i n (y) c o s h (y)

∵ U_{x} \neq V_{y}, U_{y} \neq - V_{x}

∴ f (z) i s d o n t s a t i s f y c a u c h y r i e m a n c o n d i t i o n

⟨ M . T ⟩

Commented by Tawa11 last updated on 01/Sep/21

God bless you sir . I appreciate your time

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