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Homework Statement
show that the function f(z)=zRe(z) is only differentiable at the origin.
im completely lost with is, its probably very easy.. but don't know how to start.
f(x+iy)=x+iy*Re(z)=x(x+iy)? idk
Proving f(z)=zRe(z) is Differentiable at Origin
- Thread starter fredrick08
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show that the function f(z)=zRe(z) is only differentiable at the origin.
im completely lost with is, its probably very easy.. but don't know how to start.
f(x+iy)=x+iy*Re(z)=x(x+iy)? idk
- #2
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I believe you need to use the Cauchy-Riemann equations, u_x = v_y, v_x = -u_y, where
u is the real part of the function and v is the imaginary part, and u_x is the derivative of u with respect to x etc.
As you stated, the function can be written as x(x+iy) = x^2 + ixy, so u = x^2, v = xy. You will find the functions only satisfy the C-R equations at the origin.
u is the real part of the function and v is the imaginary part, and u_x is the derivative of u with respect to x etc.
As you stated, the function can be written as x(x+iy) = x^2 + ixy, so u = x^2, v = xy. You will find the functions only satisfy the C-R equations at the origin.
- #3
gabbagabbahey
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fredrick08 said:
So far so good, now what are the real and imaginary parts, [itex]u(x,y)[/itex] and [itex]v(x,y)[/itex], of this expression? What conditions must the partial derivatives of [itex]u(x,y)[/itex] and [itex]v(x,y)[/itex] satisfy for f(x+iy) to be differentiable?
EDIT: NT123 beat me to it.
