Cauchy-Riemann equations (or condition)
Cauchy-Riemann equations Proof of Cauchy-Riemann Equation
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Liouville's Theorem in Complex Analysis
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Liouville's Theorem Art-15. Liouville's Theorem This theorem is very important, if you read this theorem carefully then some of the theorems coming up are based on this on which you can apply it and there are many questions where you can apply this theorem and you can solve those questions with Liouville's Theorem . Statement: If a function \( f(z) \) is analytic for all finite values of \( z \) and is bounded, then \( f(z) \) is constant. Proof: Let \( z_1, z_2 \) be any two points of the \( z \)-plane. Draw a circle \( C \) with center at the origin and radius \( R \), enclosing the points \( z_1 \) and \( z_2 \) so that: \( |z_1| < R \) and \( |z_2| < R \) Since \( f(z) \) is bounded, there exists \( M > 0 \) such that \( |f(z)| \leq M \) on \( C \). By Cauchy's integral formula, we have \( f(z_1) = \frac{1}{2\pi i} \int_C \frac{f(z)}{z - z_1} \, dz \) and \( f(...
Intersection of Two Subspaces is a Subspace but no need of Union
Intersection and Union of Subspaces Intersection of Two Subspaces is a Subspace Proof: Let \( S \) and \( T \) be two subspaces of a vector space \( V \) over a field \( F \). We shall prove that \( S \cap T \) is also a subspace of \( V \). Let \( \alpha, \beta \in F \) and \( x, y \in S \cap T \). Then \( x, y \in S \) and \( x, y \in T \). Since \( S \) and \( T \) are subspaces of \( V \), we have: \[ \alpha x + \beta y \in S \] \[ \alpha x + \beta y \in T \] Therefore, \[ \alpha x + \beta y \in S \cap T \] Thus, \( \alpha x + \beta y \in S \cap T \) for all \( \alpha, \beta \in F \) and \( x, y \in S \cap T \). Hence, \( S \cap T \) is a subspace of \( V \). Union of Two Subspaces Need Not Be a Subspace Now, by giving an example we shall prove that the union of two subspaces need not be a subspace. Example: Let \( V = \mathbb{R}^3 = \{(x, y, z) : x, y, z \in \mathbb{R}\} \) be a vector space over the field \( \mathbb{R} \) of all rea...
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