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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,
∂u/∂x=∂v/∂y and ∂u/∂y= -∂v/∂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) =½ In(x² +y²) and v(x, y) = tanֿ¹(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 ∂²u/∂x² + ∂²u/∂y²=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) = x² +y²
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

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Solution Summary

The solution contains explanation for the cauchy-reimann condition and its illustration for different functions.
Calculation of the gradient of a function and proving that it is normal to the level curve of the function.

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