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.

Algebra & Trigonometry with Analytic Geometry
13th Edition
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Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
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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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