Is the Function ( f(z) z^2 ) Differentiable?

Is the Function ( f(z) z^2 ) Differentiable?

In the realm of complex analysis, the differentiability of a function is a fundamental concept. Specifically, we explore whether the function ( f(z) z^2 ) is differentiable at various points in the complex plane. To delve into this, we will examine the limit behavior of the difference quotient as ( Delta z ) tends to zero.

Introduction to the Function

Consider the real-valued function ( w f(z) z^2 ). To determine the differentiability of this function, we need to evaluate the limit of the difference quotient ( frac{Delta w}{Delta z} ) as ( Delta z ) approaches zero.

Derivation and Limit Calculation

Let's start by expressing the difference quotient ( frac{Delta w}{Delta z} ) in terms of ( z ) and ( Delta z ). We begin with the expression:

[ frac{Delta w}{Delta z} frac{z (Delta z)^2 - z^2}{Delta z} ]

Next, we factor out ( z ) and simplify the expression:

[ frac{Delta w}{Delta z} frac{z Delta z bar{z} bar{Delta z} - z bar{z}}{Delta z} bar{z} bar{Delta z} z frac{bar{Delta z}}{Delta z} ]

This brings us to the expression given in Equation (1):

[ frac{Delta w}{Delta z} bar{z} bar{Delta z} z frac{bar{Delta z}}{Delta z} ]

Now, let's analyze the limit of ( frac{Delta w}{Delta z} ) as ( Delta z ) approaches zero. To do this, we will consider two particular cases: horizontal and vertical approaches.

Horizontal and Vertical Approaches

Horizontal Approach (Real Axis):

If we let ( Delta z ) approach zero along the real axis, then ( Delta z Delta x ) and ( bar{Delta z} Delta z ). Thus, the expression simplifies to:

[ frac{Delta w}{Delta z} bar{z} Delta z z frac{bar{Delta z}}{Delta z} bar{z} Delta z z ]

Vertical Approach (Imaginary Axis):

On the other hand, if ( Delta z ) approaches zero along the imaginary axis, we have ( Delta z i Delta y ) and ( bar{Delta z} -i Delta y ). Therefore:

[ frac{Delta w}{Delta z} bar{z} (-i Delta y) z frac{-i Delta y}{i Delta y} bar{z} - Delta z - z ]

Uniqueness of Limits

For the limit to exist, the results of both the horizontal and vertical approaches must be the same. Specifically, we need:

[ bar{z} Delta z z bar{z} - Delta z - z ]

By simplifying and solving for ( z ), we get:

[ bar{z} Delta z z bar{z} - Delta z - z Rightarrow bar{z} Delta z z bar{z} Delta z z 0 Rightarrow (bar{z} z 1) (Delta z) 0 ]

This implies that either ( Delta z 0 ) or ( bar{z} z 1 0 ). Since ( Delta z ) can approach zero in many different ways, we focus on the second case:

[ bar{z} z 1 0 Rightarrow |z|^2 1 0 Rightarrow |z|^2 -1 Rightarrow z 0 ]

Therefore, the limit of ( frac{Delta w}{Delta z} ) exists only when ( z 0 ).

Conclusion

In summary, the function ( f(z) z^2 ) is differentiable only at ( z 0 ). At this point, the derivative is given by:

[ frac{dw}{dz} 0 ]

This result underscores the importance of the behavior of the function at specific points in complex analysis, reflecting the unique properties of differentiability in the complex domain.

For further exploration, one may consider other complex functions and their differentiability, including their applications in various fields such as engineering and physics.