Solution 6.

See text and/or instructor's solution manual.

Answer.   Let    be fixed, then the circle      in the z-plane

is mapped onto the cardioid      in the w-plane.

Solution.   The equation of a circle in the -plane with center is given in polar coordinates    by the formula

.

For the mapping    we have  ,  and  .

Use the substitutions    and   in    to obtain

.

Square both sides and get

.

Interesting Fact.

To obtain the standard form of a cardioid use the trigonometric identity  ,  and write

Now make the substitution   and get the standard form of a cardioid:

.

Furthermore, the transformation    maps the point      onto the point   .

For the function     we have      and   .

Since   ,   the mapping is not conformal at  ,  but  .

Hence, by Theorem 10.2 in Section 10.1 the mapping    magnifies angles at    by the factor  .

The rays tangent to the circle at    make an angle    and  ,  respectively.

The image rays tangent to the two curves forming the cusp at    make the angles    and  ,  respectively.

Since these angles point in the same direction, the cusp in the cardioid forms an angle of  0°.

We are done.

Aside.  We can let Mathematica double check our work.

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Aside.  We can look at the equations.

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We are really done.

Aside.  We can explore some graphs.

The circle      with center  .

Rays tangent to the circle at the origin make an angle    and  .

Remark.  The image of these two rays will be a single ray making an angle  .

The cardioid    .

The ray tangent to the two curves forming the cusp at the origin makes an angle  .

A "zoom in" of the curve   .

The ray tangent to the two curves forming the cusp at the origin makes an angle  .

We are really really done.

Aside.  We can re-label the center of the circle and the angle of the tangent line as shown in Figure 11.62.

The circle      with center  .

Rays tangent to the circle at the origin make an angle    and  .

Remark.  The image of these two rays will be a single ray making an angle  .

The cardioid    .

The ray tangent to the two curves forming the cusp at the origin makes an angle  .

A "zoom in" of the curve   .

The ray tangent to the two curves forming the cusp at the origin makes an angle  .

Again, An Interesting Fact.

To obtain the standard form of a cardioid use the trigonometric identity  ,  and write

Now make the substitution   and get the standard form of a cardioid:

.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell