Dictionary of Conformal Mapping
Part III

Common Mappings

Example 26.  The conformal mapping    .
It can be constructed via the Schwarz-Christoffel integral  ,
also  .
The image of the upper half-plane    is the infinite strip    slit along the horizontal ray  .
Remark. This is Exercise 5 in Section 11.9, and is illustrated in Figure 11.79.
Hint: Set    and  , and the angles are .

Details 26.

Example 27.  The conformal mapping  .
It can be constructed via the Schwarz-Christoffel integral  ,
also  .
The image of the upper half-plane    is the semi-infinite strip  , and the angles are .
Remark. Hint: Set    and  , and the angles are .

Details 27.

Example 28.  The conformal mapping   .
It can be constructed via the Schwarz-Christoffel integral  .
The image of the upper half-plane    is the three quadrants IV, I, II slit along the segment  .
Remark. This is Exercise 11 (Extra Example 4) in Section 11.9 and Exercise 4 in Section 11.10, and is illustrated in Figure 11.85 and Figure 11.91.
Hint: Set    and  , and the angles are .

Details 28.

Example 29.  The conformal mapping  .
It can be constructed via the Schwarz-Christoffel integral  .
The image of the upper half-plane    is the upper half-plane slit along the segment from  .
Remark. This is Exercise 9 in Section 11.9 and Exercise 3 (ExtraExample 3) in  Section 11.10, and is illustrated in Figure 11.83 and  Figure 11.90.
Hint: Set  .  In the w-plane , and the exterior angles are  .

Details 29.

Example 30.  The conformal mapping  .
It can be constructed via the Schwarz-Christoffel integral  ,
also  .
The image of the upper half-plane    is the exterior of the semi-infinite strip  .
Remark. This is Exercise 6 in Section 11.9 and Exercise 2 (Extra Example 2) in  Section 11.10, and is illustrated in Figure 11.80 and Figure 11.89.
Hint: Set  .  In the w-plane , and the exterior angles are  .

Details 30.

Example 31.  The conformal mapping  .
It can be constructed via the Schwarz-Christoffel integral  ,
also  .
The image of the upper half-plane    is the portion of the first quadrant    that lies below the region where  ,
the right angle channel in the first quadrant, which is bounded by the coordinate axes and the rays  .
Remark. This is Example 11.28 in Section 11.9 and Exercise 9 (Extra Example 3) in Section 11.1, and is illustrated in Figure 11.74 and Figure 11.107.
Hint: Set ,  ,  ,  ,  and  ,  ,  ,  ,
and the angles are .

Example 32.  The conformal mapping   .
It can be constructed via the Schwarz-Christoffel integral  ,
also  .
The image of the upper half-plane    is the region in the -plane bounded by the negative  -axis,  the segment ,  and the ray  .
Remark. This is Exercise 3 (Extra Example 1) in Section 11.9 and Example 11.29 in  Section 11.10, and is illustrated in Figure 11.77 and Figure 11.87.
Hint: Set    and   ,  and the angles are .

Details 32.

Example 33.  The conformal mapping   .
It can be constructed via the Schwarz-Christoffel integral  ,
also  .
The image of the upper half-plane    is the upper half-plane together with the semi-infinite strip -1<u<1, v<0.
Remark. This is Exercise 4 (Extra Example 2) in Section 11.9 and Example 11.34 in Section 11.11, and is illustrated in Figure 11.78. and Figure 11.98.
Hint: Set    and  , where  ,  ,    and   ,  ,  ,  and the angles are  .

Details 33.

Example 34.  The conformal mapping   .
It can be constructed via the Schwarz-Christoffel integral  ,
also  .
The image of the upper half-plane    is the portion of the upper half plane that lies outside the quadrant .
Remark. This is Exercise 8 in Section 11.9 and Exercise 7 (Extra Example 1) in Section 11.11, and is illustrated in Figure 11.82 and Figure Figure 11.105.
Hint: Set   and , and the angles are  .

Details 34.

Example 35.  The conformal mapping   .
It can be constructed via the Schwarz-Christoffel integral  ,
also  .
The image of the upper half-plane    is the portion of the upper half plane that lies outside the region bounded by the rays and   .
Remark. This is Exercise 10 (Extra Example 3) in Section 11.9 and Exercise 8 (Extra Example 2) in Section 11.11, and is illustrated in Figure 11.84 and Figure 11.106.
Hint: Set   and ,  and the angles are  .
Use the change of variable    in the resulting integral.

Details 35.

Example 36.  The conformal mapping   .
It can be constructed via the Schwarz-Christoffel integral  ,
also  .
The image of the upper half-plane    is first quadrant slit along the ray  .
Remark. This is Exercise 15 in Section 11.9 and Exercise 10 (Extra Example 4) in Section 11.11, and is illustrated in Figure 11.86 and Figure 11.108.
Hint:  Hint: Set    and  . , and the angles are  .

Details 36.

Example 37.  The conformal mapping  .
It can be constructed via the Schwarz-Christoffel integral  ,
also,  w = .
The image of the upper half-plane    is the portion of the upper half plane     that lies above the segment    and the ray  .
Remark. This is Exercise 5 (Extra Example 5) in  Section 11.10, and is illustrated in Figure 11.92.
Hint:  Set  .  In the w-plane , and the exterior angles are  .

Details 37.

Chapter 2. Complex Functions
1. Complex Functions and Linear Mappings
2. The Mappings and
3. Complex Limits and Continuity
4. Branches of Complex Functions
5. The Reciprocal Transformation

Chapter 5. Elementary Functions

6. The Complex Exponential Function
7. The Complex Logarithm Function
8. Complex Exponents and Powers
9. Trigonometric and Hyperbolic Functions
10. Inverse Trigonometric and Hyperbolic Functions

Chapter 10. Conformal Mapping

11. Basic Properties of Conformal Mappings
12. Mobius Transformations - Bilinear Transformations
13. Mappings Involving Elementary Functions
14. Mappings by Trigonometric Functions

Chapter 11. Applications of Harmonic Functions

15. Preliminaries
16. Invariance of Laplace's Equation and the Dirichlet Problem
17. Poisson's Integral Formula for the Upper Half Plane
18. Two-Dimensional Mathematical Models
20. Two-Dimensional Electrostatics
21. Two-Dimensional Fluid Flow
22. The Joukowski Airfoil
23. The Schwarz-Christoffel Transformation
24. Image of a Fluid Flow
25. Sources and Sinks

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