### Mathematical Methods Lecture 14 of 34

October 18, 2011 by K.S. Narain

K.S. Narain , ICTP

Some review of Characteristic Polynomials was said. Note that in order to be more specific for this semester several things in chapter one have not been mentioned just like tensor calculation and so on. You could easily find most of them in tutorial . The definition of characteristic polynomials by taking a n*n matrix and imposing nth degree polynomial in $\lambda$ is given. Some examples are also given. Defining some subspaces with some special examples have been re-expressed. Decomposing several vectors as a linear combination of the matrices realized. Making some subspaces which consist of generalized eigen vectors with rank less than r is used. Taking basis for any vector by taking into account generalized eigen vectors has been used. We proved that generalized igen vectors from different subspaces are linearly independent (was given in the class). We present two examples on how to define degeneracy on a 3*3 Matrix. We talked a little bit about hamiltonian theorem. Cauchy-Rieman condition and Binomial expansion and Integration is given. Introducing branch cut the necessary conditions for Cauchy-Rieman by continuity of partial derivative are mentioned. Binomial expansion was explained . Also talk about Taylor expansion. We give an example of any function which satisfies Laplace equation which called Harmonic function and in particular we give example in 2 dimensions. The definition of integration with respect to z is given. Section 9 Conformal transformation is not given but we follow the rest of the chapter. We give an example of having anaytic function at some point which must be differentiable at that point as well as the other point. We introduce branch cut by applying Cauchy theorem.

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