Question Number 152771 by Tawa11 last updated on 01/Sep/21
$$\mathrm{If}\:\:\:\:\mathrm{f}\left(\mathrm{z}\right)\:\:\:=\:\:\:\:\mathrm{z}\:\mathrm{sin}\left(\mathrm{z}\right)\:\:\:+\:\:\:\mid\mathrm{z}\mid^{\mathrm{2}} ,\:\:\:\:\:\:\:\mathrm{verify}\:\mathrm{if}\:\:\:\mathrm{f}\left(\mathrm{z}\right)\:\:\:\mathrm{satisfy}\:\mathrm{cauchy}\:\mathrm{rieman} \\ $$$$\mathrm{condition} \\ $$
Commented by alisiao last updated on 01/Sep/21
![f(z)= z sin(z) + z z^_ f(z) = (x + iy ) sin(x+iy) + x^2 −y^2 f(z)= (x +iy)[ sin(x) cos(iy) + cos(x) sin(iy)] + x^2 −y^2 f(z)=(x+iy) [ sin(x) cosh(y) + i cos(x) sinh(y)]+x^2 −y^2 f(z)= (x sin(x)cosh(y) − y cos(x)sinh(y) +x^2 −y^2 ] + i [ y sin(x) cosh(y) +x cos(x)sinh(y)] U(x,y) = x sin(x) cosh(y) − y cos(x)sinh(y) +x^2 −y^2 V (x,y) = y sin(x)cosh(y) + x cos(y)sinh(y) U_x = x cos(x) cosh(y) +sin(x) cosh(y) + y sin(x) sinh(y) + 2x U_y = y sin(x) sinh(y)+sin(x) cosh(y) − y cos(x) cosh(y) − cos(x) sinh(y) − 2y V_x = y cos(x) cosh(y) + cos(y) sinh(y) V_y = y sin(x) sinh(y)+sin(x) cosh(y)+x cos(y) cosh(y) − x sin(y) cosh (y) ∵ U_x ≠ V _y , U_y ≠ − V_x ∴ f(z) is dont satisfy cauchy rieman condition ⟨ M . T ⟩](https://www.tinkutara.com/question/Q152780.png)
$${f}\left({z}\right)=\:{z}\:{sin}\left({z}\right)\:+\:{z}\:\overset{\_} {{z}} \\ $$$$ \\ $$$${f}\left({z}\right)\:=\:\left({x}\:+\:{iy}\:\right)\:{sin}\left({x}+{iy}\right)\:+\:{x}^{\mathrm{2}} −{y}^{\mathrm{2}} \\ $$$$ \\ $$$${f}\left({z}\right)=\:\left({x}\:+{iy}\right)\left[\:{sin}\left({x}\right)\:{cos}\left({iy}\right)\:+\:{cos}\left({x}\right)\:{sin}\left({iy}\right)\right]\:+\:{x}^{\mathrm{2}} −{y}^{\mathrm{2}} \\ $$$$ \\ $$$${f}\left({z}\right)=\left({x}+{iy}\right)\:\left[\:\:{sin}\left({x}\right)\:{cosh}\left({y}\right)\:+\:{i}\:{cos}\left({x}\right)\:{sinh}\left({y}\right)\right]+{x}^{\mathrm{2}} −{y}^{\mathrm{2}} \\ $$$$ \\ $$$${f}\left({z}\right)=\:\left({x}\:{sin}\left({x}\right){cosh}\left({y}\right)\:−\:{y}\:{cos}\left({x}\right){sinh}\left({y}\right)\:+{x}^{\mathrm{2}} −{y}^{\mathrm{2}} \:\right]\:+\:{i}\:\left[\:{y}\:{sin}\left({x}\right)\:{cosh}\left({y}\right)\:+{x}\:{cos}\left({x}\right){sinh}\left({y}\right)\right] \\ $$$$ \\ $$$${U}\left({x},{y}\right)\:=\:{x}\:{sin}\left({x}\right)\:{cosh}\left({y}\right)\:−\:{y}\:{cos}\left({x}\right){sinh}\left({y}\right)\:+{x}^{\mathrm{2}} −{y}^{\mathrm{2}} \\ $$$$ \\ $$$${V}\:\left({x},{y}\right)\:=\:{y}\:{sin}\left({x}\right){cosh}\left({y}\right)\:+\:{x}\:{cos}\left({y}\right){sinh}\left({y}\right) \\ $$$$ \\ $$$${U}_{{x}} \:=\:{x}\:{cos}\left({x}\right)\:{cosh}\left({y}\right)\:+{sin}\left({x}\right)\:{cosh}\left({y}\right)\:+\:{y}\:{sin}\left({x}\right)\:{sinh}\left({y}\right)\:+\:\mathrm{2}{x} \\ $$$$ \\ $$$${U}_{{y}} \:=\:{y}\:{sin}\left({x}\right)\:{sinh}\left({y}\right)+{sin}\left({x}\right)\:{cosh}\left({y}\right)\:−\:{y}\:{cos}\left({x}\right)\:{cosh}\left({y}\right)\:−\:{cos}\left({x}\right)\:{sinh}\left({y}\right)\:−\:\mathrm{2}{y} \\ $$$$ \\ $$$${V}_{{x}} \:=\:{y}\:{cos}\left({x}\right)\:{cosh}\left({y}\right)\:+\:{cos}\left({y}\right)\:{sinh}\left({y}\right) \\ $$$$ \\ $$$${V}_{{y}} \:=\:{y}\:{sin}\left({x}\right)\:{sinh}\left({y}\right)+{sin}\left({x}\right)\:{cosh}\left({y}\right)+{x}\:{cos}\left({y}\right)\:{cosh}\left({y}\right)\:−\:{x}\:{sin}\left({y}\right)\:{cosh}\:\left({y}\right) \\ $$$$ \\ $$$$\because\:{U}_{{x}} \:\neq\:{V}\:_{{y}} \:\:\:\:\:\:\:\:,\:{U}_{{y}} \:\neq\:−\:{V}_{{x}} \\ $$$$ \\ $$$$\therefore\:{f}\left({z}\right)\:{is}\:{dont}\:{satisfy}\:{cauchy}\:{rieman}\:{condition} \\ $$$$ \\ $$$$\langle\:{M}\:.\:{T}\:\rangle \\ $$
Commented by Tawa11 last updated on 01/Sep/21
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{I}\:\mathrm{appreciate}\:\mathrm{your}\:\mathrm{time} \\ $$