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Cauchy-Riemann Equations

Use the Cauchy-Riemann Equations to verify the analyticity of the following functions, check where they are differentiable, and find their derivative.

\( \mathsf{a)}\;\; \displaystyle w=\left(x^{2}-y^{2}-2x\right)+2iy\left(x-1\right) \)

\[ \mathsf{a)}\;\; \displaystyle w=\left(x^{2}-y^{2}-2x\right)+2iy\left(x-1\right) \]

Solution

\( \mathsf{b)}\;\; \displaystyle w=(\operatorname{e}^{y}+\operatorname{e}^{-y})\sin x+i(\operatorname{e}^{y}+\operatorname{e}^{-y})\cos x \)

\[ \mathsf{b)}\;\; \displaystyle w=(\operatorname{e}^{y}+\operatorname{e}^{-y})\sin x+i(\operatorname{e}^{y}+\operatorname{e}^{-y})\cos x \]

Solution

c) \( \displaystyle w=\frac{x}{x^{2}+y^{2}}-i\frac{y}{x^{2}+y^{2}} \)

Solution

d) \( \displaystyle w=x^{2}y^{2}+i2x^{2}y^{2} \)

Solution

e) \( \displaystyle w=\operatorname{e}^{y}(\cos x+i\sin x) \)

Solution

Use the Cauchy-Riemann Equations to verify the analyticity of the following functions, check where they are differentiable, and find their derivative.

a) \( \displaystyle w=\bar{z} \)

Solution

b) \( \displaystyle w=\text{Re}\;z \)

Solution

c) \( \displaystyle w=z\;\text{Im}\;z \)

Solution

d) \( \displaystyle w=z\;\text{Re}\;z \)

Solution

e) \( \displaystyle w=|z| \)

Solution

f) \( \displaystyle w=|\;z-1\;|^{2} \)

Solution

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