- #1
fredrick08
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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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In summary, the conversation discusses the function f(z)=zRe(z) and how to prove that it is only differentiable at the origin. The participants mention using the Cauchy-Riemann equations and determining the real and imaginary parts of the function. They also mention that the C-R equations are only satisfied at the origin.
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
NT123
- 28
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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.
What is the definition of differentiability at a point?
The concept of differentiability at a point in complex analysis is similar to that in real analysis. A function f is said to be differentiable at a point z0 if the limit of the difference quotient (f(z)-f(z0))/(z-z0) exists as z approaches z0. If this limit exists, then f is said to be differentiable at z0.
How do we prove differentiability at the origin for a complex function?
To prove differentiability at the origin for a complex function f(z), we need to show that the limit of the difference quotient (f(z)-f(0))/z as z approaches 0 exists. This can be done by expressing f(z) in terms of its real and imaginary parts, and then showing that the limit of the difference quotient for each part exists. Alternatively, we can use the Cauchy-Riemann equations to show that the partial derivatives at the origin exist and are equal.
What is the significance of proving differentiability at the origin?
Proving differentiability at the origin is important because it implies that the function is differentiable at every point in a neighborhood of the origin. This means that the function is smooth and well-behaved in that neighborhood, and we can use techniques from calculus to analyze its behavior.
What are the conditions for proving differentiability at the origin for a function of a single variable?
In order to prove differentiability at the origin for a function of a single variable, we need the function to be continuous at the origin and have a well-defined derivative at the origin. This means that the left and right limits of the difference quotient must exist and be equal at the origin.
Can we use the same approach to prove differentiability at the origin for functions of several variables?
Yes, the approach for proving differentiability at the origin for functions of several variables is similar to that for functions of a single variable. We need the function to be continuous at the origin and have well-defined partial derivatives at the origin. If the partial derivatives exist and are equal at the origin, then the function is differentiable at the origin.
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