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# Cauchy Reimann condition and analytic functions

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1. a) The Cauchy-Riemann equation is the name given to the following pair of equations,
âˆ‚u/âˆ‚x=âˆ‚v/âˆ‚y and âˆ‚u/âˆ‚y= -âˆ‚v/âˆ‚x which connects the partial derivatives of two functions u(x, y) and v(x, y)

i) if u(x, y) =e^x cosy and v(x, y) =e^x siny, how do I prove that these functions satisfy the Cauchy-Riemann equations
ii) if u(x, y) =Â½ In(xÂ² +yÂ²) and v(x, y) = tan&#1471;Â¹(y/x), how do I prove that theses functions satisfy the Cauchy-Riemann equations
iii) if u and v are any functions that satisfy the Cauchy-Riemann equations,
how do I prove that âˆ‚Â²u/âˆ‚xÂ² + âˆ‚Â²u/âˆ‚yÂ²=0

b) If f is a real valued function of two variables, the set of points (x, y) for which f(x, y)=c, for some value of the constant c, is called a level curve( or contour line) of the function. How do I illustrate the level curves for the following functions:
i) f(x, y) = xÂ² +yÂ²
ii) g(x, y) = xy

How would I calculate the gradient vectors of these functions and confirm in each case that the direction of this vector at any point is normal to the level curve passing through it