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Review of complex numbers 22=R202 2122=R₁₂+8) 01+02 41=Rje z=Re Csz-Re(+2x/8) The complex conjugate of z = Rei=a+bi is z = Re = a - bi, which is the reflection of z across the real axis. Note that Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. -2 CB-2/8 Iz²=zz Re Re = R2e0=R2 =|z|= √√zz = √√a² + b² = R. Let D and D' be simply connected domains in C, and let f: DD' be a biholomorphic (conformal) map. 1. Preservation of Laplace's Equation: Show that if u : D'→R is a harmonic function on D', then u of is a harmonic function on D. 2. Conformal Invariance of the Laplacian: Prove that the Laplacian operator is conformally invariant up to a scaling factor. Specifically, demonstrate that for f : DD' biholomorphic and u twice continuously differentiable. A(f) = f(z)²(Au) of. 3. Dirichlet Problem and Conformal Mapping: Consider the Dirichlet problem for a harmonic function on D' with boundary values given by a continuous function : OD'→R. Use the conformal map f to transform this problem to D and solve for the harmonic function on D. 4. Schwarz-Pick Lemma Extension: ⚫Extend the Schwarz-Pick Lemma to bounded symmetric domains and discuss its implications for the hyperbolic metric under biholomorphic mappings. 5. Applications to Fluid Dynamics: Apply the results from parts 1-3 to model potential flow around a simple obstacle. Specifically, use a conformal map to transform the flow domain and solve for the velocity potential and stream function.

Review of complex numbers 22=R202 2122=R₁₂+8) 01+02 41=Rje z=Re Csz-Re(+2x/8) The complex conjugate of z = Rei=a+bi is z = Re = a - bi, which is the reflection of z across the real axis. Note that Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. -2 CB-2/8 Iz²=zz Re Re = R2e0=R2 =|z|= √√zz = √√a² + b² = R. Let D and D' be simply connected domains in C, and let f: DD' be a biholomorphic (conformal) map. 1. Preservation of Laplace's Equation: Show that if u : D'→R is a harmonic function on D', then u of is a harmonic function on D. 2. Conformal Invariance of the Laplacian: Prove that the Laplacian operator is conformally invariant up to a scaling factor. Specifically, demonstrate that for f : DD' biholomorphic and u twice continuously differentiable. A(f) = f(z)²(Au) of. 3. Dirichlet Problem and Conformal Mapping: Consider the Dirichlet problem for a harmonic function on D' with boundary values given by a continuous function : OD'→R. Use the conformal map f to transform this problem to D and solve for the harmonic function on D. 4. Schwarz-Pick Lemma Extension: ⚫Extend the Schwarz-Pick Lemma to bounded symmetric domains and discuss its implications for the hyperbolic metric under biholomorphic mappings. 5. Applications to Fluid Dynamics: Apply the results from parts 1-3 to model potential flow around a simple obstacle. Specifically, use a conformal map to transform the flow domain and solve for the velocity potential and stream function.

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Complex Numbers
Section4.3: The Complex Plane
Problem 2ECP
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Review of complex numbers 22=R202 2122=R₁₂+8) 01+02 41=Rje z=Re Csz-Re(+2x/8) The complex conjugate of z = Rei=a+bi is z = Re = a - bi, which is the reflection of z across the real axis. Note that Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. -2 CB-2/8 Iz²=zz Re Re = R2e0=R2 =|z|= √√zz = √√a² + b² = R. Let D and D' be simply connected domains in C, and let f: DD' be a biholomorphic (conformal) map. 1. Preservation of Laplace's Equation: Show that if u : D'→R is a harmonic function on D', then u of is a harmonic function on D. 2. Conformal Invariance of the Laplacian: Prove that the Laplacian operator is conformally invariant up to a scaling factor. Specifically, demonstrate that for f : DD' biholomorphic and u twice continuously differentiable. A(f) = f(z)²(Au) of. 3. Dirichlet Problem and Conformal Mapping: Consider the Dirichlet problem for a harmonic function on D' with boundary values given by a continuous function : OD'→R. Use the conformal map f to transform this problem to D and solve for the harmonic function on D. 4. Schwarz-Pick Lemma Extension: ⚫Extend the Schwarz-Pick Lemma to bounded symmetric domains and discuss its implications for the hyperbolic metric under biholomorphic mappings. 5. Applications to Fluid Dynamics: Apply the results from parts 1-3 to model potential flow around a simple obstacle. Specifically, use a conformal map to transform the flow domain and solve for the velocity potential and stream function.
Transcribed Image Text:Review of complex numbers 22=R202 2122=R₁₂+8) 01+02 41=Rje z=Re Csz-Re(+2x/8) The complex conjugate of z = Rei=a+bi is z = Re = a - bi, which is the reflection of z across the real axis. Note that Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. -2 CB-2/8 Iz²=zz Re Re = R2e0=R2 =|z|= √√zz = √√a² + b² = R. Let D and D' be simply connected domains in C, and let f: DD' be a biholomorphic (conformal) map. 1. Preservation of Laplace's Equation: Show that if u : D'→R is a harmonic function on D', then u of is a harmonic function on D. 2. Conformal Invariance of the Laplacian: Prove that the Laplacian operator is conformally invariant up to a scaling factor. Specifically, demonstrate that for f : DD' biholomorphic and u twice continuously differentiable. A(f) = f(z)²(Au) of. 3. Dirichlet Problem and Conformal Mapping: Consider the Dirichlet problem for a harmonic function on D' with boundary values given by a continuous function : OD'→R. Use the conformal map f to transform this problem to D and solve for the harmonic function on D. 4. Schwarz-Pick Lemma Extension: ⚫Extend the Schwarz-Pick Lemma to bounded symmetric domains and discuss its implications for the hyperbolic metric under biholomorphic mappings. 5. Applications to Fluid Dynamics: Apply the results from parts 1-3 to model potential flow around a simple obstacle. Specifically, use a conformal map to transform the flow domain and solve for the velocity potential and stream function.
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