COMPLEX ANALYSIS: Maple Worksheets, 2001
(c) John H. Mathews Russell W. Howell
mathews@fullerton.edu howell@westmont.edu

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COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9
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CHAPTER 3 ANALYTIC and HARMONIC FUNCTIONS

Section 3.1 Differentiable Functions

Does the notion of a derivative of a complex function make sense? If so, how should it be defined, and what does it represent? These and other questions will be the focus of the next few sections.

Using our imagination, we take our lead from elementary Calculus and define the derivate of at , written f '( ), by

f '( ) = ,

provided that the limit exists. When this happens, we say that the function is differentiable at . If we write , then this definition can be expressed in the form

f '( ) = .

Example 3.1, Page 94. Use the limit definition to find the derivative of .

> df:='df': dz:='dz': f:='f': z:='z': z0:='z0':
f := z ->z^3:
`f(z) ` = f(z); ` `;
`Form the difference quotient.`;
dfdz := (f(z) - f(z0))/(z - z0):
df/dz = dfdz; ` `;

Substitution of for is indeterminate.

> subs(z=z0, dfdz);

Error, numeric exception: division by zero

The derivative is the limit of the difference quotient.

> Limit(df/dz, z=z0) = limit(dfdz, z=z0);

The difference quotient can be simplified before taking a limit.

> df/dz = dfdz;
Q := simplify(dfdz):
df/dz = Q;
`f '(z0) = `, Limit(df/dz, z=z0) = subs(z=z0, Q);

Which is known to be the formula for the derivative of .

Example 3.2, Page 94. Use the limit definition to show that is NOWHERE differentiable.

> f:='f': z:='z':
f := z -> conjugate(z):
`f(z) ` = f(z);

Form the difference quotient using a change in in .

> dx:='dx': dX:='dX': dy:='dy': dY:='dY':
X:='X': X0:='X0': Y:='Y': Y0:='Y0':
var := {dX='dx',dY='dy',X='x',X0='x0',Y='y',Y0='y0'}:
assume(dX,real); assume(dY,real);
assume(X,real); assume(Y,real);
assume(X0,real); assume(Y0,real);
dFdX := (f(X0 + dX + I*Y0) - f(X0 + I*Y0))/
(X0 + dX + I*Y0 - X0 - I*Y0):
dfdx := subs(var, dFdX):
`f(z0+dx) ` = f(z0+dx),` and `,
`f(z0) ` = f(z0); ` `;
`f(z0+dx) - f(z0)`/dx = dfdx;

And this limit is easy to compute:

> lim1 := limit(dfdx , dx=0):
Limit(df/dx, x=0) = lim1;

Form the difference quotient using a change in in .

> dFdY := (f(X0 + I*(Y0+dY)) - f(X0 + I*Y0))/
(X0 + I*(Y0+dY) - X0 - I*Y0):
dfdy := subs(var, dFdY):
`f(z0+Idy) - f(z0)`/`I dy` = dfdy;
`f(z0+Idy) - f(z0)`/`I dy` = simplify(dfdy);

And this limit is easy to compute:

> lim2 := limit(dfdy , dy=0):
Limit(df/dy, dy=0) = lim2;

The two limits are different for any point .

> Limit(df/dx, dx=0) <> Limit(df/dy, dy=0);
lim1 <> lim2;

>

Since the two limits are different for any point , does NOT exist for any .

Definition 3.1: Analytic

We say that the complex function is analytic at the point provided there is some >0 such that exists for all . In other words, must be differentiable not only at , but also at all points in some -neighborhood of .

Definition 3.1: Entire

If is analytic on the whole complex plane then said to be entire .

The rules for differentiation on page 96.

> c:='c': f:='f': F:='F': z:='z':
F := z -> c*f(z):
print(`Derivative of a scalar multiple.`);
`F(z)` = F(z);
`F '(z)` = diff(F(z), z);

> a:='a': b:='b': f:='f': g:='g': H:='H': z:='z':
H := z -> a*f(z) + b*g(z):
print(`Derivative of a linear combination multiple.`);
`H(z)` = H(z);
`H '(z)` = diff(H(z), z);

> f:='f': g:='g': P:='P': z:='z':
P := z -> f(z)*g(z):
print(`Derivative of the product of two functions.`);
`P(z)` = P(z);
`P '(z)` = diff(P(z), z);

> f:='f': F:='F': g:='g': Q:='Q': z:='z':
Q := z -> f(z)/g(z):
print(`Derivative of the quotient of two functions.`);
`Q(z)` = Q(z);
`Q '(z)` = normal(diff(Q(z), z));

> f:='f': g:='g': H:='H': z:='z':
H := z -> f(g(z)):
print(`Derivative of the composition of two functions.`);
`H(z)` = H(z);
`H '(z)` = diff(H(z), z);

Example 3.3, Page 96. Use the rules to find the derivative of ,
and then use this result to find the derivative of .

> f:='f': g:= 'g': h:='h': w:='w': z:='z':
f := z -> z^2 + I*2*z + 3:
g := z -> z^4:
h := z -> g(f(z)):
`f(z)` = f(z);
`f '(z)` = diff(f(z), z); ` `;
`g(z)` = g(z);
`g '(z)` = diff(g(z), z); ` `;
`h(z) = g(f(z))` = g(f(z));
`h '(z)` = diff(h(z), z);

Theorem 3.1 (Differentiable implies continuous)

If is differentiable at , then is continuous at .

Theorem 3.2 (L'Hopital's Rule)

Assume and are analytic at . If we have , , and , then

= .

Example about L'Hopital's Rule, Page 98.

> f:='f': g:='g': z:='z':
f := z -> z^2 + z - 1 - 3*I:
g := z -> z^2 - 2*z + 2:
`f(z) ` = f(z);
`g(z) ` = g(z); ` `;
`f(z)/g(z) ` = f(z)/g(z);

Direct substitution of is indeterminate.

> f(1+I)/g(1+I);

Error, numeric exception: division by zero

Find .

> Limit(f(z)/g(z), z=1+I) = limit(f(z)/g(z), z=1+I);

Which agrees with the L'Hopital rule computation.

> f1:='f1': g1:='g1': Z:='Z':
f1 := z -> subs(Z=z, diff(f(Z), Z)):
g1 := z -> subs(Z=z, diff(g(Z), Z)):
`f '(z)/g '(z)` = f1(z)/g1(z); ` `;
`f '(1+I)` = f1(1+I);
`g '(1+I)` = g1(1+I); ` `;
`f '(1+I)/g '(1+I)` = f1(1+I)/g1(1+I);

Theorem 3.3 (The first identity theorem)

Let be an analytic function in the domain . Suppose for all that , where is a constant.

Then constant in .

End of Section 3.1.