--- title: "Conformal Maps and Mobius Transformations" description: "See that an analytic map with nonzero derivative preserves angles, and use Mobius maps to move standard regions onto one another." canonical: https://lightmysky.com/learn/mathematics/conformal-maps-and-mobius-transformations-mt_WtHCbAT4GI source: https://lightmysky.com/learn/mathematics/conformal-maps-and-mobius-transformations-mt_WtHCbAT4GI.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Conformal Maps and Mobius Transformations See that an analytic map with nonzero derivative preserves angles, and use Mobius maps to move standard regions onto one another. Subject: Mathematics · Area: Complex Analysis · Ages 21 to 22 Page: https://lightmysky.com/learn/mathematics/conformal-maps-and-mobius-transformations-mt_WtHCbAT4GI ## Ready when they can - Explain why a holomorphic map with nonzero derivative preserves oriented angles locally - Construct the unique Mobius transformation matching corresponding entries of two ordered triples, each containing three distinct extended-plane points - Map the upper half-plane bijectively onto the unit disc and transfer harmonic Dirichlet boundary data and solutions ## Lesson: Keep angles and bend regions Let f be holomorphic near a. A tangent vector v at a maps to f'(a)v. If f'(a) is nonzero, this multiplication rotates every tangent by the same angle and scales every length by the same positive factor. It preserves oriented angles between regular curves. This is local conformality. The exponential qualifies at every finite point: its derivative e^z never vanishes. **Example.** For f(z)=alpha*z+beta, alpha != 0, the multiplier alpha scales lengths by |alpha| and turns directions by arg(alpha). Adding beta translates the image without changing tangents. Multiplication by i turns directions 90 degrees counterclockwise. To evaluate a point image, distribute and use i^2=-1. For f(z)=2iz+1, f(2-i)=4i-2i^2+1=3+4i. For f(z)=z^2, the derivative 2z vanishes only at 0. Squaring sends a ray of direction theta to direction 2*theta, so it does not preserve angles at 0. Local conformality also differs from global injectivity. The exponential is conformal everywhere, but e^(z+2*pi*i)=e^z. In particular, 0 and 2*pi*i both map to 1. A Mobius map is T(z)=(az+b)/(cz+d), where ad-bc != 0. Its derivative is T'(z)=(ad-bc)/(cz+d)^2, nonzero wherever cz+d != 0. A zero determinant instead gives a constant ratio where defined, provided the denominator is not identically zero. Multiplying all four coefficients by one nonzero number leaves the map unchanged. The extended complex plane adds one point, infinity. A Mobius map is a bijection of this plane. If c != 0, its pole -d/c maps to infinity and T(infinity)=a/c, found by a limit. If c=0, infinity maps to infinity and T is affine. Mobius maps send generalized circles to generalized circles: these are circles and lines completed by infinity. Two ordered triples of distinct extended-plane points determine exactly one Mobius map matching corresponding entries. For finite p,q,r, A(z)=((z-p)/(z-r))*((q-r)/(q-p)) sends p,q,r to 0,1,infinity in that order. Normalize the target triple with B in the corresponding order. Then T=B^-1 composed with A gives the prescribed images. Use limits when infinity occurs. Distinct targets are necessary because the map is injective. **Example.** Construct T taking (0,4,infinity) to (0,1,3). From T(0)=0, set b=0. The finite image of infinity requires c != 0; rescale to c=1. Then T(infinity)=3 gives a=3. The condition T(4)=1 gives 12/(4+d)=1, so d=8. Thus T(z)=3z/(z+8), with determinant 24 != 0. If infinity instead maps to infinity, use the affine form alpha*z+beta. Evaluate a ratio by substituting into both numerator and denominator. For R(z)=(z-2)/(z+2), R(4)=2/6=1/3. For a complex denominator, multiply both parts by its conjugate. With C(z)=(z-i)/(z+i) and real x, this gives C(x)=(x-i)^2/(x^2+1)=(x^2-1-2ix)/(x^2+1). C(z)=(z-i)/(z+i) maps H={Im z>0} bijectively onto D={|w|<1}, with C(i)=0. For z=x+iy, |C(z)|^2=[x^2+(y-1)^2]/[x^2+(y+1)^2]<1 when y>0. Solving w=C(z) gives C^-1(w)=i(1+w)/(1-w). Its imaginary part is (1-|w|^2)/|1-w|^2>0 on D. Every disc point therefore has a unique inverse in H. A real-valued harmonic u satisfies Delta u=u_xx+u_yy=0. A Dirichlet problem prescribes its boundary values. For holomorphic C, Delta(u composed with C)=|C'|^2*(Delta u) composed with C. For C:H to D, set disc data h=g composed with C^-1. Once a harmonic disc solution u with data h is known, v=u composed with C solves the half-plane problem. Apply the forward map C before u. For C(z)=(z-i)/(z+i), finite real x have |C(x)|=1 and C(x) != 1. Their images form the unit circle minus 1; C(infinity)=1 fills the gap. Continuous data on the completed real boundary must have a common limit g(infinity) as x tends to either real infinity. The transformed data then satisfy h(1)=g(infinity). A disc solution continuous on the closed disc pulls back to these boundary values. **Example.** To find g(infinity) for a rational function with equal numerator and denominator degrees, divide both by the highest power of x. Terms with negative powers tend to 0. For g(x)=(5x^2+2)/(x^2+4), rewrite it as (5+2/x^2)/(1+4/x^2). The limit is 5 at both real ends. Thus the transformed boundary data have h(1)=5. **Example.** To evaluate a pulled-back solution, compute w=C(z) first and then apply u. For u(w)=Re w and z=4i, C(4i)=3i/(5i)=3/5, so v(4i)=3/5. Real parts of holomorphic functions are harmonic. Another disc example is u(w)=Re(w^2); evaluate it by squaring C(z) before taking the real part. *(drawing: For z=4i, first compute C(4i)=3/5, then apply u(w)=Re w.)* **Recap.** A nonzero holomorphic derivative preserves local angles. Distinct ordered triples fix one Mobius map in matching order. Use the inverse for boundary data and the forward map for solutions. *This lesson, question bank and flashcards were written for this dataset.* ## Practice 17 questions on this page, each with its working shown. ## Needs first - [Complex Differentiability and the Cauchy-Riemann Equations](https://lightmysky.com/learn/mathematics/complex-differentiability-and-the-cauchy-riemann-equations-mt_nZNmZcl9Ss) ## Optional connections - [Matrix multiplication and what it represents](https://lightmysky.com/learn/mathematics/matrix-multiplication-and-what-it-represents-mt_0fJOMPlCxr) - [The Argument Principle and Rouche's Theorem](https://lightmysky.com/learn/mathematics/the-argument-principle-and-rouches-theorem-mt_J1dC428M84)