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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) = tan&#1471;¹(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

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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