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The Cauchy-Riemann Equations

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u(x,y) = \tan^{-1}(y/x); v(x,y) = -\frac{1}{2}\ln(x^2+y^2)

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Yes

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u(x,y) = x^3 - 3x^2y; v(x,y) = 3x^3 + y^3

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No

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u(x,y) = e^{x^2-y^2}; v(x,y) = 2xe^{x^2-y^2}

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No

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u(x,y) = \arcsin\left(\frac{x^2-y^2}{x^2+y^2}\right); v(x,y) = \arctan\left(\frac{2xy}{x^2-y^2}\right)

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No

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u(x,y) = \sin(x)\cos(y); v(x,y) = \cos(x)\sin(y)

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No

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u(x,y) = y\exp(-x); v(x,y) = x\exp(-y)

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No

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u(x,y) = x^2 + y^2; v(x,y) = 2xy

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No

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u(x,y) = \sin(y)e^x; v(x,y) = \cos(y)e^x

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Yes

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u(x,y) = \frac{y}{x}; v(x,y) = \ln\left(\sqrt{x^2 + y^2}\right)

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No

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u(x,y) = x^2y^2; v(x,y) = (x^2 - y^2)^2

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No

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u(x,y) = \frac{x}{x^2 + y^2}; v(x,y) = \frac{y}{x^2 + y^2}

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No

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u(x,y) = \exp(-x)\cos(y); v(x,y) = \exp(-x)\sin(y)

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Yes

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u(x,y) = e^y\cos(x); v(x,y) = e^y\sin(x)

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Yes

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u(x,y) = x^3 - 3xy^2; v(x,y) = 3x^2y - y^3

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Yes

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u(x,y) = \cos(x)\cosh(y); v(x,y) = -\sin(x)\sinh(y)

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Yes

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u(x,y) = \sin(x)\sinh(y); v(x,y) = \cos(x)\cosh(y)

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No

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u(x,y) = e^x\cos(y); v(x,y) = e^x\sin(y)

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Yes

StarStarStarStar

u(x,y) = \cos(x)\sinh(y); v(x,y) = \sin(x)\cosh(y)

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No

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u(x,y) = y^2 - x^2; v(x,y) = 2xy

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Yes

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u(x,y) = y^3 - 3x^2y; v(x,y) = 3xy^2 - x^3

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Yes

StarStarStarStar

u(x,y) = \cosh(x)\sin(y); v(x,y) = \sinh(x)\cos(y)

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Yes

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u(x,y) = \ln(x^2 + y^2); v(x,y) = \arctan\left(\frac{y}{x}\right)

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No

StarStarStarStar

u(x,y) = \ln\left(\sqrt{x^2 + y^2}\right); v(x,y) = \arctan\left(\frac{y}{x}\right)

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Yes

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u(x,y) = \sinh(x)\cos(y); v(x,y) = \cosh(x)\sin(y)

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Yes

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u(x,y) = \frac{3x^2y}{x^2+y^2}; v(x,y) = x^3 - 3xy^2

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No

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u(x,y) = \cos(x^2-y^2); v(x,y) = \sinh(2xy)

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No

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u(x,y) = \frac{x}{x^2 - y^2}; v(x,y) = \frac{y}{2xy}

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No

StarStarStarStar

u(x,y) = \sin(x^2 + y^2); v(x,y) = \cos(x^2 + y^2)

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No

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u(x,y) = y\cos(x); v(x,y) = x\sin(y)

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No

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u(x,y) = \sin(x)\exp(y); v(x,y) = -\cos(x)\exp(y)

StarStarStarStar

Yes

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