Consider the following functions:(a) f(z) = x3(1 + i) − y3(1 − i)x2 + y2, (z ̸= 0), f(0) = 0,
Question
Consider the following functions:
- (a) , (),
- f(0) = 0,
Solution
To analyze the function where , we will look at this function in terms of the complex variables and see if it has any specific properties such as continuity, differentiability, or any interesting features.
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Break Down the Problem
- The function is given in terms of its real and imaginary components.
- Identify the real part and the imaginary part of the function .
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Relevant Concepts
- A function of a complex variable is analytic if it satisfies the Cauchy-Riemann equations:
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Analysis and Detail
- First, we will express in terms of and :
- Here,
- Considering and isolating the real and imaginary parts:
- Real part
- Imaginary part
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Verify and Summarize
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Now, we calculate the partial derivatives:
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Check the Cauchy-Riemann equations:
- is not equal to in general.
- is not equal to .
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Thus, since the Cauchy-Riemann equations do not hold, is not differentiable in the complex sense in any area where .
Final Answer
The function is not analytic anywhere in .
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