Analytic Function (3+x-iy)^7 in Domain [z]<2?

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The function f(z) = (3+x-iy)^7 is not analytic in the domain |z| < 2. This is demonstrated by the calculation of the derivative with respect to the conjugate variable, yielding df/d\bar{z} = 7(3+\bar{z})^6, which does not equal zero. Therefore, the function fails the Cauchy-Riemann equations necessary for analyticity. As a result, (3+x-iy)^7 cannot be considered an analytic function in the specified domain. The analysis confirms that the function is dependent on the conjugate variable, indicating non-analytic behavior.
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(3+x-iy)^7 is analytic function of the complex variable z=x+iy in the domain [z]<2?
 
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Nope, f(z) = (3+x-iy)^7 = (3+\bar{z})^7

\frac{df}{d\bar{z}} = 7(3+\bar{z})^6 \neq 0, \,\mathrm{for}\; |z| &lt; 2.
 

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