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

$$\mathrm{If}\mathrm{f}\left(\mathrm{z}\right)=\mathrm{z}\sin \left(\mathrm{z}\right)+\mid \mathrm{z}{\mid }^2,\mathrm{verify}\mathrm{if}\mathrm{f}\left(\mathrm{z}\right)\mathrm{satisfy}\mathrm{cauchy}\mathrm{rieman}$$$$\mathrm{condition}$$

Commented by alisiao last updated on 01/Sep/21

$$f\left(z\right)=zsin\left(z\right)+z\stackrel{\mathrm{\_}}{z}$$$$$$$$f\left(z\right)=(x+iy)sin(x+iy)+x^2-y^2$$$$$$$$f\left(z\right)=(x+iy)[sin\left(x\right)cos\left(iy\right)+cos\left(x\right)sin\left(iy\right)]+x^2-y^2$$$$$$$$f\left(z\right)=(x+iy)[sin\left(x\right)cosh\left(y\right)+icos\left(x\right)sinh\left(y\right)]+x^2-y^2$$$$$$$$f\left(z\right)=(xsin\left(x\right)cosh\left(y\right)-ycos\left(x\right)sinh\left(y\right)+x^2-y^2]+i[ysin\left(x\right)cosh\left(y\right)+xcos\left(x\right)sinh\left(y\right)]$$$$$$$$U(x,y)=xsin\left(x\right)cosh\left(y\right)-ycos\left(x\right)sinh\left(y\right)+x^2-y^2$$$$$$$$V(x,y)=ysin\left(x\right)cosh\left(y\right)+xcos\left(y\right)sinh\left(y\right)$$$$$$$$U_x=xcos\left(x\right)cosh\left(y\right)+sin\left(x\right)cosh\left(y\right)+ysin\left(x\right)sinh\left(y\right)+2x$$$$$$$$U_y=ysin\left(x\right)sinh\left(y\right)+sin\left(x\right)cosh\left(y\right)-ycos\left(x\right)cosh\left(y\right)-cos\left(x\right)sinh\left(y\right)-2y$$$$$$$$V_x=ycos\left(x\right)cosh\left(y\right)+cos\left(y\right)sinh\left(y\right)$$$$$$$$V_y=ysin\left(x\right)sinh\left(y\right)+sin\left(x\right)cosh\left(y\right)+xcos\left(y\right)cosh\left(y\right)-xsin\left(y\right)cosh\left(y\right)$$$$$$$$\because U_x\ne V{}_y,U_y\ne -V_x$$$$$$$$\therefore f\left(z\right)isdontsatisfycauchyriemancondition$$$$$$$$⟨M.T⟩$$

Commented by Tawa11 last updated on 01/Sep/21

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.\mathrm{I}\mathrm{appreciate}\mathrm{your}\mathrm{time}$$

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