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Bilinear Transformation

Bilinear Transformation. Sub Name : Engineering Mathematics – I By K.VITHYA, M.Sc.,M.Phil., Assistant Professor / Mathematics. Bilinear Transformation. Complex function. f(z) onto. 1. f(z) one-to-one. 2. Bilinear Transformation. Example:. Bilinear Transformation.

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Bilinear Transformation

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  1. Bilinear Transformation Sub Name : Engineering Mathematics – I By K.VITHYA, M.Sc.,M.Phil., Assistant Professor / Mathematics

  2. Bilinear Transformation Complex function f(z) onto 1 f(z) one-to-one 2

  3. Bilinear Transformation Example:

  4. Bilinear Transformation Def: Conformal mapping Is conformal if it preserve angles 1 Orentation 2

  5. Bilinear Transformation Theorem : Let analytic function (D onto K) Conformal mapping Theorem (Riemann Mapping Theorem): Let Unit circle

  6. Bilinear Transformation Remark: A conformal mapping of a domain D onto a domain K will map the boundary of D to the boundary of K.

  7. Bilinear Transformation Linear fractional Transformation (bilinear) Is a function of the form a, b, c, d complex number and Remark: T is a conformal mapping defined on C – {-d/c}

  8. Bilinear Transformation Theorem : T maps a circle to a circle or straight line T maps a straight line to a circle or straight line circle circle circle

  9. Bilinear Transformation Theorem : Find T so that:

  10. Bilinear Transformation Theorem : Example How to find T: Find a bilinear mapping that sends the given points to the images indicated. Think of w =T(z) and solve for w in the equation

  11. Bilinear Transformation Theorem(Riemann Mapping Theorem): How to find T: Unit circle Map the right half-plane Re(z)>0 conformally onto the unit disk |w| <1 We will map boundary to boundary Select 3 points in z-plane Select 3 images in w-plane Maintain orentation (counterclockwise) As you walk on the boundary the domain is on your left.

  12. Bilinear Transformation An integral Solution of the Dirichlet Problem for a Disk Let u be a harmonic function in the unit Disk |z|<1+e slightly larger than the unit the values of u are prescribed on the boundary circle C Given u The complex function f(z) can be written as (see p322 for derivation) Remark

  13. Bilinear Transformation Solution of Dirichlet Problem by conformal mapping Solution is the real part of The solution of this problem is The real part of the function f(z)

  14. Bilinear Transformation Solution of Dirichlet Problem by conformal mapping The solution of this problem The real part of the function f(z) Example Solve the Dirichlet problem for the right half-plane

  15. Bilinear Transformation Solution of Dirichlet Problem by conformal mapping Solve the Dirichlet problem for the right half-plane Example real Real part

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