COMPLEX ANALYSIS: Maple Worksheets, 2009
(c) John H. Mathews Russell W. Howell
mathews@fullerton.edu howell@westmont.edu

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CHAPTER 3 ANALYTIC and HARMONIC FUNCTIONS

Section 3.2 The Cauchy-Riemann Equations

We saw in the last section that computing the derivative of complex functions written in a nice form such as is a rather simple task. But life is not so easy, for many times we encounter complex functions written as . For example, suppose we had

.

Is there some criterion---perhaps involving the partial derivatives for , and - - that we can use to determine whether is differentiable, and if so, to find the value of ?

The answer to this question is yes , thanks in part to the independent discovery of two important equations relating the partial derivatives of and by the French mathematician A. L. Cauchy and the German mathematician G. F. B. Riemann.

Theorem 3.4 (Cauchy-Riemann equations)

Suppose that is differentiable at the point . Then the partial derivatives of and exist at the point ( ), and

(3-15) f '( ) = ,

and

(3-16) f '( ) = .

Equating the real and imaginary parts of gives us the Cauchy-Riemann equations

(3-17) and .

Note carefully some of the implications of this theorem:

If differentiable at , then we know the Cauchy-Riemann equations (3-17) will be satisfied at , and we can use either equation (3-15) or (3-16) to evaluate f '( ).

Taking the contrapositive, if equations (3-17) are not satisfied at , then we know automatically that is not differentiable at .

On the other hand, just because equations (3-17) are satisfied at , we cannot necessarily conclude that is differentiable at .

Derivation of Theorem 3.4

> dx:='dx': dy:='dy': f:='f': U:='U': V:='V': x:='x': y:='y':
assume(x,real);
assume(y,real);
f := proc(x,y) U(x,y) + I*V(x,y) end:
`f(x + I y) = U(x,y) + I V(x,y)`; ` `;
dfdx := evalc((f(x+dx,y)-f(x,y))/dx):
df/dx = subs({x='x',y='y'}, dfdx); ` `;
L1 := limit((f(x+dx,y)-f(x,y))/dx, dx=0):
L1:=expand(L1):
dfdy := evalc((f(x,y+dy)-f(x,y))/(I*dy)):
df/dy = subs({x='x',y='y'}, dfdy);
L2 := limit((f(x,y+dy)-f(x,y))/(I*dy), dy=0):
L2:=expand(L2):
Limit(df/dx, dx=0) = subs({x='x',y='y'}, L1);
Limit(df/dy, dy=0) = subs({x='x',y='y'}, L2); ` `;
R1 := subs(I=0,expand(L1)):
R2 := subs(I=0,expand(L2)):
`The two limits must be the same.`;
subs({x='x',y='y'}, L1 = L2); ` `;
print(`The Cauchy-Riemann equations are:`);
subs({x='x',y='y'}, R1 = R2);
subs({x='x',y='y'}, subs(I=1,L1-R1) = subs(I=1,L2-R2));

These are the Cauchy-Riemann equations
and .

Example for Page 104. The function i.e.
is known to be differentiable.
Verify that its derivative satisfies the results of the Cauchy-Riemann equations.

> f:='f': U:='U': V:='V': x:='x': y:='y': z:='z':
f := z -> z^3:
`f(z) ` = f(z);
`f(x + I y) ` = f(x+I*y);
`f(x + I y) ` = evalc(f(x+I*y));
U := proc(x,y) x^3 - 3*x*y^2 end:
V := proc(x,y) 3*x^2*y - y^3 end:
`U(x,y) ` = U(x,y);
`V(x,y) ` = V(x,y); ` `;
f1 := z -> subs(Z=z,diff(f(Z),Z)):
`f '(z) ` = f1(z);
`f '(x + I y) ` = f1(x+I*y);
`f '(x + I y) ` = evalc(f1(x+I*y)); ` `;
`f '(z) = Ux(x,y) + i Vx(x,y)` =
diff(U(x,y),x) + I* diff(V(x,y),x);
`f '(z) = Vy(x,y) - i Uy(x,y)` =
diff(V(x,y),y) - I* diff(U(x,y),y);

Example 3.6, Page 105. Verify that the complex function is NOT analytic.

> f:='f': F:='F': U:='U': V:='V': x:='x': y:='y': z:='z':
f := z -> conjugate(z)^2/z:
`f(z) ` = f(z);
`f(x + I y) ` = f(x+I*y);
assume(x,real); assume(y,real);
w1 := simplify(evalc(f(x+I*y))):
`f(x + I y)` = subs({x='x',y='y'}, w1); ` `;
x:='x': y:='y':
U := proc(x,y) (x^3-3*x*y^2)/(x^2+y^2) end:
V := proc(x,y) (-3*x^2*y+y^3)/(x^2+y^2) end:
F := proc(x,y) U(x,y) + I*V(x,y) end:
`F(x,y)` = subs({x='x',y='y'}, F(x,y));
`U(x,y)` = subs({x='x',y='y'}, U(x,y));
`V(x,y)` = subs({x='x',y='y'}, V(x,y)); ` `;
Ux := simplify(diff(U(x,y), x)): `Ux(x,y)` = Ux;
Vy := simplify(diff(V(x,y), y)): `Vy(x,y)` = Vy;
Uy := simplify(diff(U(x,y), y)): `Uy(x,y)` = Uy;
Vx := simplify(diff(V(x,y), x)): `Vx(x,y)` = Vx; ` `;
print(`0 = Ux - Vy `, 0 = simplify(Ux - Vy),
0 = numer(simplify(Ux - Vy)));
print(`0 = Uy + Vx `, 0 = simplify(Uy + Vx),
0 = numer(simplify(Uy + Vx)));

Hence the function f is differentiable only when and or at the origin.

Let's check to see if the Cauchy-Riemann equations hold at ( , ) . Using the facts that
and , we compute the limits of the difference quotients.

> `Ux(0,0)` = limit((U(x,0)-0)/(x-0), x=0);
`Vy(0,0)` = limit((V(0,y)-0)/(y-0), y=0);
`Uy(0,0)` = limit((U(0,y)-0)/(y-0), y=0);
`-Vx(0,0)` = -limit((V(x,0)-0)/(x-0), x=0);

So, and .and we see that the Cauchy-Riemann
equations hold at . But is not-analytic at ( , ) because the following two limits are distinct.

> `Along the x-axis: `, Limit(df/dz, dz=0) =
limit((F(x,0)-0)/(x-0),x=0);
`Along the line x=t, y=t: `, Limit(df/dz, dz=0) =
limit((F(t,t)-0)/(t+I*t-0),t=0);

Example 3.8, Page 108. Use the Cauchy-Riemann equations to show that
is differentiable for all .

> U:='U': V:='V': x:='x': y:='y':
U := proc(x,y) exp(-y)*cos(x) end:
V := proc(x,y) exp(-y)*sin(x) end:
`U(x,y)` = U(x,y);
`V(x,y)` = V(x,y); ` `;
`Ux(x,y)` = diff(U(x,y),x);
`Vy(x,y)` = diff(V(x,y),y);
diff(U(x,y),x) = diff(V(x,y),y);
evalb(diff(U(x,y),x) = diff(V(x,y),y)); ` `;
`Uy(x,y)` = diff(U(x,y),y);
`Vx(x,y)` = diff(V(x,y),x);
diff(U(x,y),y) = - diff(V(x,y),x);
evalb(diff(U(x,y),y) = - diff(V(x,y),x)); ` `;
`f(x+iy)` = U(x,y) + I*V(x,y);
`f '(x+iy)` = diff(U(x,y),x) + I*diff(V(x,y),x);

The Cauchy-Riemann equations hold everywhere, so that
is analytic for all .

Example 3.9, Page 108. Show that the complex function
is differentiable only at points that lie on the coordinate axes.

> U:='U': V:='V': x:='x': y:='y':
U := proc(x,y) x^3 + 3*x*y^2 end:
V := proc(x,y) y^3 + 3*x^2*y end:
`U(x,y)` = U(x,y);
`V(x,y)` = V(x,y); ` `;
`Ux(x,y)` = diff(U(x,y),x);
`Vy(x,y)` = diff(V(x,y),y);
print(`0 = Ux - Vy `,
0 = diff(U(x,y),x) - diff(V(x,y),y),
evalb(0 = diff(U(x,y),x) - diff(V(x,y),y))); ` `;
`Uy(x,y)` = diff(U(x,y),y);
`Vx(x,y)` = diff(V(x,y),x);
print(`0 = Uy + Vx `,
0 = diff(U(x,y),y) + diff(V(x,y),x),
evalb(0 = diff(U(x,y),y) + diff(V(x,y),x)));

The Cauchy-Riemann equations hold only if . So that
is analytic only when or , which occurs at points that lie on the coordinate axes.

End of Section 3.2.