18.04 S18 Topic 10: Conformal Transformations

Topic 10 Notes Jeremy Orloff 10 Conformal transformations 10.1 Introduction In this topic we will look at the geometric notion of conformal maps. It will turn out that analytic functions are automatically conformal. Once we have understood the general notion, we will look at a specific family of conformal maps called fractional linear transformations and, in particular at their geometric properties. As an application we will use fractional linear transformations to solve the Dirichlet problem for harmonic functions on the unit disk with specified values on the unit circle. At the end we will return to some questions of fluid flow. 10.2 Geometric definition of conformal mappings We start with a somewhat hand-wavy definition: Informal definition. Conformal maps are functions on � that preserve the angles between curves. More precisely: Suppose � (�) is differentiable at �0 and �(�) is a smooth curve through �0 . To be concrete, let’s suppose �(�0 ) = �0 . The function maps the point �0 to �0 = � (�0 ) and the curve � to �̃ (�) = � (�(�)). Under this map, the tangent vector � ′ (�0 ) at �0 is mapped to the tangent vector �̃ ′ (�0 ) = (� ◦�)′ (�0 ) at �0 . With these notations we have the following definition. Definition. The function � (�) is conformal at �0 if there is an angle � and a scale � > 0 such that for any smooth curve �(�) through �0 the map � rotates the tangent vector at �0 by � and scales it by �. That is, for any �, the tangent vector (� ◦�)′ (�0 ) is found by rotating � ′ (�0 ) by � and scaling it by �. If � (�) is defined on a region �, we say it is a conformal map on � if it is conformal at each point � in �. Note. The scale factor � and rotation angle � depends on the point �, but not on any of the curves through �. Example 10.1. The figure below shows a conformal map � (�) mapping two curves through �0 to two curves through �0 = � (�0 ). The tangent vectors to each of the original curves are both rotated and scaled by the same amount. 1 10 CONFORMAL TRANSFORMATIONS 2 y f (γ1 (t)) v z 7→ w = f (z) γ2 (t) (f ◦ γ2 )′ (t0 ) (f ◦ γ1 )′ (t0 ) f (γ2 (t)) w0 = f (z0 ) γ2′ (t0 ) γ1 (t) z0 γ1′ (t0 ) w0 x u A conformal map rotates and scales all tangent vectors at �0 by the same ammount. Remark 1. Conformality is a local phenomenon. At a different point �1 the rotation angle and scale factor might be different. Remark 2. Since rotations preserve the angles between vectors, a key property of conformal maps is that they preserve the angles between curves. Example 10.2. Recall that way back in Topic 1 we saw that � (�) = �2 maps horizontal and vertical grid lines to mutually orthogonal parabolas. We will see that � (�) is conformal. So, the orthogonality of the parabolas is no accident. The conformal map preserves the right angles between the grid lines. v y z 7→ w = z 2 x 10.3 u Tangent vectors as complex numbers In 18.02, you used parametrized curves �(�) = (�(�), �(�)) in the ��-plane. Considered this way, the tangent vector is just the derivative: � ′ (�) = (�′ (�), �′ (�)). Note, as a vector, (�′ , �′ ) represents a displacement. If the vector starts at the origin, then the endpoint is at (�′ , �′ ). More typically we draw the vector starting at the point �(�). 10 CONFORMAL TRANSFORMATIONS 3 In 18.04, we use parametrized curves �(�) = �(�) + ��(�) in the complex plane. Considered this way, the tangent vector is just the derivative: � ′ (�) = �′ (�) + ��′ (�). It should be clear that these representations are equivalent. The vector (�′ , �′ ) and the complex number �′ + ��′ both represent the same displacement. Also, the length of a vector and the angle between two vectors is the same in both representations. Thinking of tangent vectors to curves as complex numbers allows us to recast conformality in terms of complex numbers. Theorem 10.3. If � (�) is conformal at �0 then there is a complex number � = �e�� such that the map � multiplies tangent vectors at �0 by �. Conversely, if the map � multiplies all tangent vectors at �0 by � = �e�� then � is conformal at �0 . Proof. By definition � is conformal at �0 means that there is an angle � and a scalar � > 0 such that the map � rotates tangent vectors at �0 by � and scales them by �. This is exactly the effect of multiplication by � = �e�� . 10.4 Analytic functions are conformal Theorem 10.4. (Operational definition of conformal) If � is analytic on the region � and � ′ (�0 ) ≠ 0, then � is conformal at �0 . Furthermore, the map � multiplies tangent vectors at �0 by � ′ (�0 ). Proof. The proof is a quick computation. Suppose � = �(�) is curve through �0 with �(�0 ) = �0 . The curve �(�) is transformed by � to the curve � = � (�(�)). By the chain rule we have �� (�(�)) || = � ′ (�(�0 ))� ′ (�0 ) = � ′ (�0 )� ′ (�0 ). �� ||�0 The theorem now follows from Theorem 10.3. Example 10.5. (Basic example) Suppose � = �e�� and consider the map � (�) = ��. Geometrically, this map rotates every point by � and scales it by �. Therefore, it must have the same effect on all tangent vectors to curves. Indeed, � is analytic and � ′ (�) = � is constant. Example 10.6. Let � (�) = �2 . So � ′ (�) = 2�. Thus the map � has a different affect on tangent vectors at different points �1 and �2 . Example 10.7. (Linear approximation) Suppose � (�) is analytic at � = 0. The linear approximation (first two terms of the Taylor series) is � (�) ≈ � (0) + � ′ (0)�. If �(�) is a curve with �(�0 ) = 0 then, near �0 , � (�(�)) ≈ � (0) + � ′ (0)�(�). That is, near 0, � looks like our basic example plus a shift by � (0). Example 10.8. The map � (�) = � has lots of nice geometric properties, but it is not conformal. It preserves the length of tangent vectors and the angle between tangent vectors. The reason it isn’t conformal is that is does not rotate tangent vectors. Instead, it reflects them across the �-axis. In other words, it reverses the orientation of a pair of vectors. Our definition of conformal maps requires that it preserves orientation. 10 CONFORMAL TRANSFORMATIONS 10.5 4 Digression to harmonic functions Theorem 10.9. If � and � are harmonic conjugates and � = � + �� has � ′ (�0 ) ≠ 0, then the level curves of � and � through �0 are orthogonal. Note. We proved this in an earlier topic using the Cauchy-Riemann equations. Here will make an argument involving conformal maps. Proof. First we’ll examine how � maps the level curve �(�, �) = �. Since � = � + ��, the image of the level curve is � = � + ��, i.e it’s (contained in) a vertical line in the �-plane. Likewise, the level curve �(�, �) = � is mapped to the horizontal line � = � + ��. Thus, the images of the two level curves are orthogonal. Since � is conformal it preserves the angle between the level curves, so they must be orthogonal. y v z 7→ w = g(z) = u(x, y) + iv(x, y) w0 = g(z0 ) u=a v=b u=a v=b z0 w0 x � = � + �� maps level curves of � and � to grid lines. 10.6 Riemann mapping theorem The Riemann mapping theorem is a major theorem on conformal maps. The proof is fairly technical and we will skip it. In practice, we will write down explicit conformal maps between regions. Theorem 10.10. (Riemann mapping theorem) If � is simply connected and not the whole plane, then there is a bijective conformal map from � to the unit disk. Corollary. For any two such regions there is a bijective conformal map from one to the other. We say they are conformally equivalent. 10.7 Fractional linear transformations Definition. A fractional linear transformation is a function of the form � (�) = �� + � , where �, �, �, � are complex constants and �� − �� ≠ 0 �� + � These are also called Mobius transforms or bilinear transforms. We will abbreviate fractional linear transformation as FLT. Simple point. If �� − �� = 0 then � (�) is a constant function. u 10 CONFORMAL TRANSFORMATIONS 5 Proof. The full proof requires that we deal with all the cases where some of the coefficients are 0. We’ll give the proof assuming � ≠ 0 and leave the case � = 0 to you. Assuming � ≠ 0, the condition �� − �� = 0 implies � (�, �) = (�, �). � So, (�∕�)(�� + �) � � (�) = = . �� + � � That is, � (�) is constant. Extension to ∞. It will be convenient to consider linear transformations to be defined on the extended complex plane � ∪ {∞} by defining { �∕� if � ≠ 0 � (∞) = ∞ if � = 0 � (−�∕�) = ∞ if � ≠ 0. 10.7.1 Examples Example 10.11. (Scale and rotate) Let � (�) = ��. If � = � is real this scales the plane. If � = e�� it rotates the plane. If � = �e�� it does both at once. y v 2i 2i z 7→ w = az i i a = reiθ x −2 −1 −i 1 2 u −2 −1 −i 1 2 Multiplication by � = �e�� scales by � and rotates by � Note that � is the fractional linear transformation with coefficients [ ] [ ] � � � 0 = . � � 0 1 (We’ll see below the benefit of presenting the coefficients in matrix form!) Example 10.12. (Scale and rotate and translate) Let � (�) = �� + �. Adding the � term introduces a translation to the previous example. y v 2i z 7→ w = az + b i −2 −1 −i 2i i x 1 2 −2 −1 −i u 1 2 The map � = �� + � scales, rotates and shifts the square. 10 CONFORMAL TRANSFORMATIONS 6 Note that � is the fractional linear transformation with coefficients [ ] [ ] � � � � = . � � 0 1 Example 10.13. (Inversion) Let � (�) = 1∕�. This is called an inversion. It turns the unit circle inside out. Note that � (0) = ∞ and � (∞) = 0. In the figure below the circle that is outside the unit circle in the � plane is inside the unit circle in the � plane and vice-versa. Note that the arrows on the curves are reversed. y v z 7→ w = 1/z i i x 1 1 u The map � = 1∕� inverts the plane. Note that � is the fractional linear transformation with coefficients [ ] [ ] � � 0 1 = . � � 1 0 Example 10.14. Let �−� . �+� We claim that this maps the �-axis to the unit circle and the upper half-plane to the unit disk. � (�) = Proof. First take � real, then √ |� − � | �2 + 1 |� (�)| = =√ = 1. |� + �| �2 + 1 So, � maps the �-axis to the unit circle. Next take � = � + �� with � > 0, i.e. � in the upper half-plane. Clearly |� + 1| > |� − 1|, so |� + �| = |� + �(� + 1)| > |� + �(� − 1)| = |� − �|, implying that |� (�)| = |� − �| < 1. |� + �| So, � maps the upper half-plane to the unit disk. We will use this map frequently, so for the record we note that � (�) = 0, � (∞) = 1, � (−1) = �, � (0) = −1, � (1) = −�. 10 CONFORMAL TRANSFORMATIONS 7 These computations show that the real axis is mapped counterclockwise around the unit circle starting at 1 and coming back to 1. z 7→ w = y z−i z+i v i i -1 0 x 1 1 u −i The map � = �−� maps the upper-half plane to the unit disk. �+� 10.7.2 Lines and circles Theorem. A linear fractional transformation maps lines and circles to lines and circles. Before proving this, note that it does not say lines are mapped to lines and circles to circles. For example, in Example 10.14 the real axis is mapped the unit circle. You can also check that inversion � = 1∕� maps the line � = 1 + �� to the circle |� − 1∕2| = 1∕2. Proof. We start by showing that inversion maps lines and circles to lines and circles. Given � and � = 1∕� we define �, �, � and � by � = � + �� and �= � − �� 1 = 2 = � + �� � � + �2 So, � � and �=− 2 . � 2 + �2 � + �2 Now, every circle or line can be described by the equation �= �� + �� + �(�2 + �2 ) = � (If � = 0 it descibes a line, otherwise a circle.) We convert this to an equation in �, � as follows. ⇔ ⇔ �� + �� + �(�2 + �2 ) = � �� �� � + 2 +� = 2 2 2 2 � +� � +� � + �2 2 2 �� − �� + � = �(� + � ). In the last step we used the fact that �2 + �2 = |�|2 = 1∕|�|2 = 1∕(�2 + �2 ). We have shown that a line or circle in �, � is transformed to a line or circle in �, �. This shows that inversion maps lines and circles to lines and circles. We note that for the inversion � = 1∕�. 10 CONFORMAL TRANSFORMATIONS 8 1. Any line not through the origin is mapped to a circle through the origin. 2. Any line through the origin is mapped to a line through the origin. 3. Any circle not through the origin is mapped to a circle not through the origin. 4. Any circle through the origin is mapped to a line not through the origin. Now, to prove that an arbitrary fractional linear transformation maps lines and circles to lines and circles, we factor it into a sequence of simpler transformations. First suppose that � = 0. So, � (�) = (�� + �)∕�. Since this is just translation, scaling and rotating, it is clear it maps circles to circles and lines to lines. Now suppose that � ≠ 0. Then, � (�� + �) + � − �� + � � (�) = = � �� + � �� + � �� � = � � − ��∕� + � �� + � So, � = � (�) can be computed as a composition of transforms � → �1 = �� + � → �2 = 1∕�1 → �= � + (� − ��∕�)�2 � We know that each of the transforms in this sequence maps lines and circles to lines and circles. Therefore the entire sequence does also. □ 10.7.3 Mapping �� to �� It turns out that for two sets of three points �1 , �2 , �3 and �1 , �2 , �3 there is a fractional linear transformation that takes �� to �� . We can construct this map as follows. Let �1 (�) = (� − �1 )(�2 − �3 ) . (� − �3 )(�2 − �1 ) Notice that �1 (�1 ) = 0, Likewise let �2 (�) = �1 (�2 ) = 1, �1 (�3 ) = ∞. (� − �1 )(�2 − �3 ) . (� − �3 )(�2 − �1 ) Notice that �2 (�1 ) = 0, �2 (�2 ) = 1, Now � (�) = �2−1 ◦�1 (�) is the required map. �2 (�3 ) = ∞. 10 CONFORMAL TRANSFORMATIONS 9 10.7.4 Correspondence with matrices We can identify the transformation � (�) = �� + � �� + � [ ] � � . � � with the matrix This identification is useful because of the following algebraic facts. [ ] [ ] � � � � 1. If � ≠ 0 then and � correspond to the same FLT. � � � � Proof. This follows from the obvious equality �� + � ��� + �� = . �� + � ��� + �� [ ] [ ] � � � � 2. If � (�) corresponds to � = and �(�) corresponds to � = then composition � � � ℎ � ◦�(�) corresponds to matrix multiplication ��. Proof. The proof is just a bit of algebra. ( ) � ((�� + � )∕(�� + ℎ)) + � �� + � (�� + ��)� + �� + �ℎ � ◦�(�) = � = = �� + ℎ � ((�� + � )∕(�� + ℎ)) + � (�� + ��)� + �� + �ℎ [ ][ ] [ ] � � � � �� + �� �� + �ℎ �� = = � � � ℎ �� + �� �� + �ℎ The claimed correspondence is clear from the last entries in the two lines above. [ ] � � 3. If � (�) corresponds to � = then � has an inverse and � −1 (�) corresponds to �−1 and � � [ ] � −� also to , i.e. to �−1 without the factor of 1∕ det(�). −� � Proof. Since ��−1 = � it is clear from the previous fact that � −1 corresponds to �−1 . Since [ ] � −� 1 −1 � = �� − �� −� � [ ] � −� −1 Fact 1 implies � and both correspond to the same FLT, i.e. to � −1 . −� � Example 10.15. [ ] � � 1. The matrix corresponds to � (�) = �� + �. 0 1 [ �� ] e 0 2. The matrix corresponds to rotation by 2�. 0 e−�� [ ] 0 1 3. The matrix corresponds to the inversion � = 1∕�. 1 0 10 10.8 CONFORMAL TRANSFORMATIONS 10 Reflection and symmetry 10.8.1 Reflection and symmetry in a line Example 10.16. Suppose we have a line � and a point �1 not on �. The reflection of �1 in � is the point �2 so that � is the perpendicular bisector to the line segment �1 �2 . Since there is exactly one such point �2 , the reflection of a point in a line is unique. Definition. If �2 is the reflection of �1 in �, we say that �1 and �2 are symmetric with respect to the line �. In the figure below the points �1 and �2 are symmetric in the �-axis. The points �3 and �4 are symmetric in the line �. y z3 S b z1 b a z4 x a z2 In order to define the reflection of a point in a circle we need to work a little harder. Looking back at the previous example we can show the following. Fact. If �1 and �2 are symmetric in the line �, then any circle through �1 and �2 intersects � orthogonally. Proof. Call the circle �. Since � is the perpendicular bisector of a chord of �, the center of � lies on �. Therefore � is a radial line, i.e. it intersects � orthogonally. y z3 z1 a S b b z4 x a z2 Circles through symmetric points intersect the line at right angles. 10 CONFORMAL TRANSFORMATIONS 11 10.8.2 Reflection and symmetry in a circle We will adapt this for our definition of reflection in a circle. So that the logic flows correctly we need to start with the definition of symmetric pairs of points. Definition. Suppose � is a line or circle. A pair of points �1 , �2 is called symmetric with respect to � if every line or circle through the two points intersects � orthogonally. First we state an almost trivial fact. Fact. Fractional linear transformations preserve symmetry. That is, if �1 and �2 are symmetric in a line or circle �, then, for an FLT � , � (�1 ) and � (�2 ) are symmetric in � (�). Proof. The definition of symmetry is in terms of lines and circles, and angles. Fractional linear transformations map lines and circles to lines and circles and, being conformal, preserve angles. □ Theorem. Suppose � is a line or circle and �1 a point not on �. There is a unique point �2 such that the pair �1 , �2 is symmetric in �. Proof. Let � be a fractional linear transformation that maps � to a line. We know that �1 = � (�1 ) has a unique reflection �2 in this line. Since � −1 preserves symmetry, �1 and �2 = � −1 (�2 ) are symmetric in �. Since �2 is the unique point symmetric to �1 the same is true for �2 vis-a-vis �1 . This is all shown in the figure below. w = T (z) y y S T (S) z1 z2 = T −1 (w2 ) x x w1 = T (z1 ) w2 We can now define reflection in a circle. Definition. The point �2 in the theorem is called the reflection of �1 in �. 10.8.3 Reflection in the unit circle Using the symmetry preserving feature of fractional linear transformations, we start with a line and transform to the circle. Let � be the real axis and � the unit circle. We know the FLT �−� � (�) = �+� maps � to �. We also know that the points � and � are symmetric in �. Therefore �1 = � (�) = �−� �+� and �2 = � (�) = �−� �+� are symmetric in �. Looking at the formulas, it is clear that �2 = 1∕�1 . This is important enough that we highlight it as a theorem. Theorem. (Reflection in the unit circle) The reflection of � = � + �� = �e�� in the unit circle is � + �� 1 � e�� = = = . � � |�|2 �2 + �2 10 CONFORMAL TRANSFORMATIONS 12 The calculations from 1∕� are all trivial. Notes. 1. It is possible, but more tedious and less insightful, to arrive at this theorem by direct calculation. 2. If � is on the unit circle then 1∕� = �. That is, � is its own reflection in the unit circle –as it should be. 3. The center of the circle 0 is symmetric to the point at ∞. The figure below shows three pairs of points symmetric in the unit circle: 1 �1 = 2; �1 = , 2 �2 = 1 + �; �2 = 1+� , 2 �3 = −2 + �; �3 = −2 + � . 5 y i z3 z2 w2 w3 w1 1 z1 x Pairs of points �� ; �� symmetric in the unit circle. Example 10.17. Reflection in the circle of radius �. Suppose � is the circle |�| = � and �1 is a point not on �. Find the reflection of �1 in �. Solution: Our strategy is to map � to the unit circle, find the reflection and then map the unit circle back to �. Start with the map � (�) = � = �∕�. Clearly � maps � to the unit circle and �1 = � (�1 ) = �1 ∕�. The reflection of �1 is �2 = 1∕�1 = �∕�1 . Mapping back from the unit circle by � −1 we have �2 = � −1 (�2 ) = ��2 = �2 ∕�1 . Therefore the reflection of �1 is �2 ∕�1 . 10 CONFORMAL TRANSFORMATIONS 13 Here are three pairs of points symmetric in the circle of radius 2. Note, that this is the same figure as the one above with everything doubled. �1 = 4; �1 = 1, �2 = 2 + 2�; �2 = 1 + �, �3 = −4 + 2�; �3 = −4 + 2� . 5 y z3 z2 i w2 1 w1 w3 z1 x Pairs of points �� ; �� symmetric in the circle of radius 2. Example 10.18. Find the reflection of �1 in the circle of radius � centered at �. Solution: Let � (�) = (� − �)∕�. � maps the circle centered at � to the unit circle. The inverse map is � −1 (�) = �� + �. So, the reflection of �1 is given by mapping � to � (�), reflecting this in the unit circle, and mapping back to the original geometry with � −1 . That is, the reflection �2 is �1 → �1 − � � �2 → → �2 = + �. � �1 − � �1 − � We can now record the following important fact. Fact. (Reflection of the center) For a circle � with center � the pair �, ∞ is symmetric with respect to the circle. Proof. This is an immediate consequence of the formula for the reflection of a point in a circle. For example, the reflection of � in the unit circle is 1∕�. So, the reflection of 0 is infinity. Example 10.19. Show that if a circle and a line don’t intersect then there is a pair of points �1 , �2 that is symmetric with respect to both the line and circle. Solution: By shifting, scaling and rotating we can find a fractional linear transformation � that maps the circle and line to the following configuration: The circle is mapped to the unit circle and the line to the vertical line � = � > 1. y 1/r 1 a r x 10 CONFORMAL TRANSFORMATIONS 14 For any real �, �1 = � and �2 = 1∕� are symmetric in the unit circle. We can choose a specific � so that � and 1∕� are equidistant from �, i.e. also symmetric in the line � = �. It is clear geometrically that this can be done. Algebraically we solve the equation � + 1∕� =� 2 ⇒ Thus �1 = � −1 (� + �2 − 2�� + 1 = 0 ⇒ �=�+ √ �2 − 1 ⇒ √ 1 = � − �2 − 1. � √ √ �2 − 1) and �2 = � −1 (� − �2 − 1) are the required points. Example 10.20. Show that if two circles don’t intersect then there is a pair of points �1 , �2 that is symmetric with respect to both circles. Solution: Using a fractional linear transformation that maps one of the circles to a line (and the other to a circle) we can reduce the problem to that in the previous example. Example 10.21. Show that any two circles that don’t intersect can be mapped conformally to concentric circles. Solution: Call the circles �1 and �2 . Using the previous example start with a pair of points �1 , �2 which are symmetric in both circles. Next, pick a fractional linear transformation � that maps �1 to 0 and �2 to infinity. For example, � − �1 � (�) = . � − �2 Since � preserves symmetry 0 and ∞ are symmetric in the circle � (�1 ). This implies that 0 is the center of � (�1 ). Likewise 0 is the center of � (�2 ). Thus, � (�1 ) and � (�2 ) are concentric. 10.9 Solving the Dirichlet problem for harmonic functions In general, a Dirichlet problem in a region � asks you to solve a partial differential equation in � where the values of the solution on the boundary of � are specificed. Example 10.22. Find a function � harmonic on the unit disk such that { 1 for 0 < � < � �(e�� ) = 0 for −� < � < 0 This is a Dirichlet problem because the values of � on the boundary are specified. The partial differential equation is implied by requiring that � be harmonic, i.e. we require ∇2 � = 0. We will solve this problem in due course. 10.9.1 Harmonic functions on the upper half-plane Our strategy will be to solve the Dirichlet problem for harmonic functions on the upper half-plane and then transfer these solutions to other domains. Example 10.23. Find a harmonic function �(�, �) on the upper half-plane that satisfies the boundary condition { 1 for � < 0 �(�, 0) = 0 for � > 0 10 CONFORMAL TRANSFORMATIONS 15 Solution: We can write down a solution explicitly as 1 �, � where � is the argument of � = � + ��. Since we are only working on the upper half-plane we can take any convenient branch with branch cut in the lower half-plane, say −�∕2 < � < 3�∕2. �(�, �) = y z = x + iy u=1 θ r1 u=0 x r2 To show � is truly a solution, we have to verify two things: 1. � satisfies the boundary conditions 2. � is harmonic. Both of these are straightforward. First, look at the point �2 on the positive �-axis. This has argument � = 0, so �(�2 , 0) = 0. Likewise arg(�1 ) = �, so �(�1 , 0) = 1. Thus, we have shown point (1). To see point (2) remember that log(�) = log(�) + ��. So, ( ) 1 log(�) . �� Since it is the real part of an analytic function, � is harmonic. � = Re Example 10.24. Suppose �1 < �2 < �3 . Find a harmonic function � on the upper half-plane that satisfies the boundary condition ⎧� ⎪ 0 ⎪� �(�, 0) = ⎨ 1 ⎪�2 ⎪� ⎩ 3 for � < �1 for �1 < � < �2 for �2 < � < �3 for �3 < � Solution: We mimic the previous example and write down the solution � � � �(�, �) = �3 + (�2 − �3 ) 3 + (�1 − �2 ) 2 + (�0 − �1 ) 1 . � � � Here, the �� are the angles shown in the figure. One again, we chose a branch of � that has 0 < � < � for points in the upper half-plane. (For example the branch −�∕2 < � < 3�∕2.) y z = x + iy u = c0 r0 θ1 x1 u = c1 r1 u = c2 θ2 x2 r2 θ3 x3 u = c3 r3 x 10 CONFORMAL TRANSFORMATIONS 16 To convince yourself that � satisfies the boundary condition test a few points: • At �3 : all the �� = 0. So, �(�3 , 0) = �3 as required. • At �2 : �1 = �2 = 0, �3 = �. So, �(�2 , 0) = �3 + �2 − �3 = �2 as required. • Likewise, at �1 and �0 , � have the correct values. As before, � is harmonic because it is the real part of the analytic function Φ(�) = �3 + (�2 − �3 ) (� − �0 ) (� − �2 ) log(� − �3 ) + 1 log(� − �2 ) + 1 log(� − �1 ). �� �� �� 10.9.2 Harmonic functions on the unit disk Let’s try to solve a problem similar to the one in Example 10.22. Example 10.25. Find a function � harmonic on the unit disk such that { 1 for −�∕2 < � < �∕2 �(e�� ) = 0 for �∕2 < � < 3�∕2 y i u=0 z u=1 x −i Solution: Our strategy is to start with a conformal map � from the upper half-plane to the unit disk. We can use this map to pull the problem back to the upper half-plane. We solve it there and then push the solution back to the disk. Let’s call the disk �, the upper half-plane �. Let � be the variable on � and � the variable on �. Back in Example 10.14 we found a map from � to �. The map and its inverse are � = � (�) = �−� , �+� w = T −1 (z) = Im(z) u=0 i z � = � −1 (�) = �� + � . −� + 1 iz+i −z+1 Im(w) u=1 ϕ=1 Re(z) w θ2 1 θ1 −1 ϕ=1 ϕ=0 −i z = T (w) = w−i w+i The function � on � is transformed by � to a function � on �. The relationships are �(�) = �◦� −1 (�) or �(�) = �◦� (�) Re(w) 10 CONFORMAL TRANSFORMATIONS 17 These relationships determine the boundary values of � from those we were given for �. We compute: � −1 (−�) = 1, � −1 (�) = −1, � −1 (1) = ∞, � −1 (−1) = 0. This shows the left hand semicircle bounding � is mapped to the segment [−1, 1] on the real axis. Likewise, the right hand semicircle maps to the two half-lines shown. (Literally, to the ‘segment’ 1 to ∞ to −1.) We know how to solve the problem for a harmonic function � on �: ( ) 1 1 1 1 �(�) = 1 − �2 + �1 = Re 1 − log(� − 1) + log(� + 1) . � � �� �� Transforming this back to the disk we have ( ) 1 1 �(�) = �◦� −1 (�) = Re 1 − log(� −1 (�) − 1) + log(� −1 (�) + 1) . �� �� If we wanted to, we could simplify this somewhat using the formula for � −1 . 10.10 Flows around cylinders 10.10.1 Milne-Thomson circle theorem The Milne-Thomson theorem allows us to insert a circle into a two-dimensional flow and see how the flow adjusts. First we’ll state and prove the theorem. Theorem. (Milne-Thomson circle theorem) If � (�) is a complex potential with all its singularities outside |�| = � then ( ) �2 Φ(�) = � (�) + � � is a complex potential with streamline on |�| = � and the same singularities as � in the region |�| > �. Proof. First note that �2 ∕� is the reflection of � in the circle |�| = �. Next we need to see that � (�2 ∕�) is analytic for |�| > �. By assumption � (�) is analytic for |�| ≤ �, so it can be expressed as a Taylor series � (�) = �0 + �1 � + �2 �2 + … Therefore, (1) ( ) ( 2 )2 �2 �2 � � = �0 + �1 + �2 +… (2) � � � All the singularities of � are outside |�| = �, so the Taylor series in Equation 1 converges for |�| ≤ �. This means the Laurent series in Equation 2 converges for |�| ≥ �. That is, � (�2 ∕�) is analytic for |�| ≥ �, i.e. it introduces no singularies to Φ(�) outside |�| = �. The last thing to show is that |�| = � is a streamline for Φ(�). This follows because for � = �e�� Φ(�e�� ) = � (�e�� ) + � (�e�� ) is real. Therefore �(�e�� ) = Im(Φ(�e�� ) = 0. 10 CONFORMAL TRANSFORMATIONS 18 10.10.2 Examples Think of � (�) as representing flow, possibly with sources or vortices outside |�| = �. Then Φ(�) represents the new flow when a circular obstacle is placed in the flow. Here are a few examples. Example 10.26. (Uniform flow around a circle) We know from Topic 6 that � (�) = � is the complex potential for uniform flow to the right. So, Φ(�) = � + �2 ∕� is the potential for uniform flow around a circle of radius � centered at the origin. Uniform flow around a circle Just because they look nice, the figure includes streamlines inside the circle. These don’t interact with the flow outside the circle. Note, that as � gets large flow looks uniform. We can see this analytically because Φ′ (�) = 1 − �2 ∕�2 goes to 1 as � gets large. (Recall that the velocity field is (�� , �� ), where Φ = � + �� . . . ) Example 10.27. (Source flow around a circle) Here the source is at � = −2 (outside the unit circle) with complex potential � (�) = log(� + 2). With the appropriate branch cut the singularities of � are also outside |�| = 1. So we can apply Milne-Thomson and obtain ( ) 1 Φ(�) = log(� + 2) + log +2 � Source flow around a circle We know that far from the origin the flow should look the same as a flow with just a source at � = −2. Let’s see this analytically. First we state a useful fact: 10 CONFORMAL TRANSFORMATIONS 19 Useful fact. If �(�) is analytic then so is ℎ(�) = �(�) and ℎ′ (�) = � ′ (�). Proof. Use the Taylor series for � to get the Taylor series for ℎ and then compare ℎ′ (�) and � ′ (�). □ Using this we have 1 1 − � + 2 �(1 + 2�) For large � the second term decays much faster than the first, so Φ′ (�) = Φ′ (�) ≈ 1 . �+2 That is, far from � = 0, the velocity field looks just like the velocity field for � (�), i.e. the velocity field of a source at � = −2. Example 10.28. (Transforming flows) If we use �(�) = �2 we can transform a flow from the upper half-plane to the first quadrant Source flow around a quarter circular corner 10.11 Examples of conformal maps and excercises As we’ve seen, once we have flows or harmonic functions on one region, we can use conformal maps to map them to other regions. In this section we will offer a number of conformal maps between various regions. By chaining these together along with scaling, rotating and shifting we can build a large library of conformal maps. Of course there are many many others that we will not touch on. For convenience, in this section we will let �0 (�) = �−� . �+� This is our standard map of taking the upper half-plane to the unit disk. Example 10.29. Let �� be the half-plane above the line � = tan(�)�, 10 CONFORMAL TRANSFORMATIONS 20 i.e., {(�, �) ∶ � > tan(�)�}. Find an FLT from �� to the unit disk. Solution: We do this in two steps. First use the rotation �−� (�) = e−�� � to map �� to the upper half-plane. Follow this with the map �0 . So our map is �0 ◦�−� (�). You supply the picture Example 10.30. Let � be the channel 0 ≤ � ≤ � in the ��-plane. Find a conformal map from � to the upper half-plane. Solution: The map � (�) = e� does the trick. (See the Topic 1 notes!) You supply the picture: horizontal lines get mapped to rays from the origin and vertical segments in the channel get mapped to semicircles. Example 10.31. Let � be the upper half of the unit disk. Show that �0−1 maps � to the second quadrant. Solution: You supply the argument and figure. Example 10.32. Let � be the upper half of the unit disk. Find a conformal map from � to the upper half-plane. Solution: The map �0−1 (�) maps � to the second quadrant. Then multiplying by −� maps this to the first quadrant. Then squaring maps this to the upper half-plane. In the end we have ( ( ))2 �� + � � (�) = −� . −� + 1 You supply the sequence of pictures. Example 10.33. Let � be the infinite well {(�, �) ∶ � ≤ 0, 0 ≤ � ≤ �}. Find a conformal map from � to the upper half-plane. � �� � � Solution: The map � (�) = e� maps � to the upper half of the unit disk. Then we can use the map from Example 10.32 to map the half-disk to the upper half-plane. You supply the sequence of pictures. Example 10.34. Show that the function � (�) = � + 1∕� maps the region shown below to the upper half-plane. 10 CONFORMAL TRANSFORMATIONS 21 � � −1 Solution: You supply the argument and figures 1 �