Theorem 3.3 (Cauchy-Riemann Equations).  Suppose that

(3-14)              ,

is differentiable at the point  .   Then the partial derivatives of    exist at the point  ,

and can be used to calculate the derivative at .  That is,

(3-14)              ,

and also

(3-15)              .

Equating the real and imaginary parts of Equations (3-14) and (3-15) gives the so-called Cauchy-Riemann Equations:

(3-16)                   and     .

Exploration for the Cauchy-Riemann Equations.

Aside.  Both and can assist us in calculating limits.

Aside.  The Mathematica solution uses the command.

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Looking at the above limits, and equating the real and imaginary parts we have the following equations.

,

and

.

In Mathematica the syntax for partial derivatives can be explained as follows.

In the expression  ,  the superscript    means, take one derivative with respect to the first variable .

In the expression  ,  the superscript    means, take one derivative with respect to the second variable .

Similarly, the expressions    and    are the    and    partial derivatives, respectively.

Therefore, we see that Mathematica can establish the Cauchy-Riemann equations

and     .

We are done.

Aside.  The Maple commands are similar.

>

>

>

Looking at the above limits, and equating the real and imaginary parts we have the following equations.

,    and

.

In Maple the syntax for partial derivatives can be explained as follows.

In the expression  ,  the subscript    means, take one derivative with respect to the first variable .

In the expression  ,  the subscript    means, take one derivative with respect to the second variable .

Similarly, the expressions    and    are the    and    partial derivatives, respectively.

Therefore, we see that Maple can establish the Cauchy-Riemann equations

and     .

Remark.  In this book the use of computers is optional.

Hopefully this text will promote their use and understanding.

(c) 2011 John H. Mathews, Russell W. Howell