To demonstrate that the function $f(z) = z^2$ is not conformal at $z = 0$, we need to show that it either:

1. Does not preserve angles at $z = 0$, or
2. Is not holomorphic with a non-zero derivative at $z = 0$.

A function is conformal at a point if it preserves angles and the magnitude of the derivative is non-zero. Let's go through the analysis for $f(z) = z^2$ at $z = 0$.

Step-by-Step Solution

1. Check Holomorphic Condition:

   The function $f(z)$ is holomorphic (analytic) if its derivative exists and is continuous. Let's compute the derivative:

   $f(z) = z^2$

   $f'(z) = \frac{d}{dz} (z^2) = 2z$

   The derivative $f'(z) = 2z$ is clearly continuous everywhere, including at $z = 0$, so the function is holomorphic everywhere in the complex plane, including at $z = 0$.

2. Check for Non-Zero Derivative at $z = 0$:

   We evaluate the derivative at $z = 0$:

   $f'(0) = 2 \cdot 0 = 0$

   Since the derivative is zero at $z = 0$, $f(z)$ fails to be conformal at this point because a conformal map must have a non-zero derivative at the point of interest. The zero derivative means that the map $f(z) = z^2$ does not locally behave like a rotation or dilation at $z = 0$. Instead, it "flattens out," failing to preserve angles.

3. Angle Preservation Condition:

   Conformality also requires that the function preserves angles between curves intersecting at a point. For $f(z) = z^2$, the map doubles the angle between curves because of the nature of squaring complex numbers. However, at $z = 0$, the derivative is zero, meaning the infinitesimal shape around this point collapses to a single point, losing any information about the angle between curves. Thus, angles are not preserved at $z = 0$.

Conclusion

Since $f'(0) = 0$, the function $f(z) = z^2$ is not conformal at $z = 0$. This is due to the derivative being zero at that point, resulting in the loss of angle preservation.

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Would you like more details on any specific part of this explanation or have any other questions?

Here are some related questions you might find interesting:

1. What are the conditions for a function to be conformal?
2. How does the Cauchy-Riemann equation relate to conformal mappings?
3. Can a function be holomorphic but not conformal?
4. What is the geometric interpretation of a conformal map?
5. How does the concept of conformality relate to isometries?
6. What are some examples of conformal maps in complex analysis?
7. How does the derivative of a complex function affect its conformality?
8. What role does the argument of a complex number play in conformal mappings?

Tip: To check if a function is conformal at a point, always verify if the derivative is non-zero at that point and whether the function is holomorphic.