2. Define (a) Bounded set. (b) Closed set. परिभाषित कीजिए : (क) परिबंध समुच्चय। (ख) संवृत समुच्घय। 3. Prove that $$ f(z)=z^{2} $$ is uniformly continuous in the domain |z।

Question

2. Define (a) Bounded set. (b) Closed set. परिभाषित कीजिए : (क) परिबंध समुच्चय। (ख) संवृत समुच्घय। 3. Prove that $$ f(z)=z^{2} $$ is uniformly continuous in the domain |z। < 1. सिद्ध कीजिए कि प्रान्त $$ |z|<1 $$ में $$ f(z)=z^{2} $$ एकसमान संवत है। 4. Prove that the series $$ e^{z}=1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\ldots $$ is absolutely and uniformly convergent. सिद्ध करो कि श्रेणी $$ e^{z}=1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\ldots $$ निरेक्ष तथा एकसमान अभिसारी है। 5. Define following maps: (a) Translation. (b) Rotation. (c) Magnification.

UpStudy AI · Accepted Answer

Let's address each part of your request step by step. ### 2. Definitions (a) **Bounded Set**: A set $$ S $$ in a metric space is called bounded if there exists a real number $$ M $$ such that the distance between any two points in $$ S $$ is less than $$ M $$. In other words, there exists a point $$ p $$ in the space such that the entire set $$ S $$ lies within a ball of radius $$ M $$ centered at $$ p $$. (b) **Closed Set**: A set $$ S $$ in a metric space is called closed if it contains all its limit points. This means that if a sequence of points in $$ S $$ converges to a point $$ x $$, then $$ x $$ must also be in $$ S $$. Alternatively, a set is closed if its complement is open. ### 3. Proving Uniform Continuity of $$ f(z) = z^2 $$ in the Domain $$ |z| < 1 $$ To prove that $$ f(z) = z^2 $$ is uniformly continuous in the domain $$ |z| < 1 $$, we need to show that for every $$ \epsilon > 0 $$, there exists a $$ \delta > 0 $$ such that for all $$ z_1, z_2 $$ in the domain, if $$ |z_1 - z_2| < \delta $$, then $$ |f(z_1) - f(z_2)| < \epsilon $$. 1. Start with the expression for the difference: $$ |f(z_1) - f(z_2)| = |z_1^2 - z_2^2| = |(z_1 - z_2)(z_1 + z_2| $$ 2. Since $$ |z_1| < 1 $$ and $$ |z_2| < 1 $$, we have: $$ |z_1 + z_2| \leq |z_1| + |z_2| < 1 + 1 = 2 $$ 3. Therefore, we can bound the difference: $$ |f(z_1) - f(z_2)| \leq 2 |z_1 - z_2| $$ 4. To ensure $$ |f(z_1) - f(z_2)| < \epsilon $$, we can choose $$ \delta = \frac{\epsilon}{2} $$. Thus, if $$ |z_1 - z_2| < \delta $$, we have: $$ |f(z_1) - f(z_2)| < 2 \delta = \epsilon $$ This shows that $$ f(z) = z^2 $$ is uniformly continuous in the domain $$ |z| < 1 $$. ### 4. Proving Absolute and Uniform Convergence of the Series $$ e^z $$ To prove that the series $$ e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \ldots $$ is absolutely and uniformly convergent, we can use the Weierstrass M-test. 1. Consider the general term of the series: $$ a_n(z) = \frac{z^n}{n!} $$ 2. We need to find a bound $$ M_n $$ such that $$ |a_n(z)| \leq M_n $$ for all $$ z $$ in a compact subset of the complex plane. 3. For $$ |z| \leq R $$ (where $$ R $$ is a positive constant), we have: $$ |a_n(z)| = \left| \frac{z^n}{n!} \right| \leq \frac{R^n}{n!} $$ 4. The series $$ \sum_{n=0}^{\infty} \frac{R^n}{n!} $$ converges (it is the Taylor series for $$ e^R $$). Therefore, we can choose $$ M_n = \frac{R^n}{n!} $$. 5. By the Weierstrass M-test, since $$ \sum_{n=0}^{\infty} M_n $$ converges, the series $$ e^z $$ converges uniformly on compact subsets of the complex plane. Thus, the series $$ e^z $$ is absolutely and uniformly convergent. ### 5. Definitions of Maps (a) **Translation**: A translation is a function that shifts every point in a space by a fixed vector. Mathematically, if $$ T $$ is a translation by a vector $$ v $$, then for any point $$ z $$ in the space, the translated point is given by: $$ T(z) = z + v $$ (b) **Rotation**: A rotation is a function that rotates every point in a space around a fixed point (the center of rotation) by a certain angle. If $$ R $$ is a rotation around the origin by an angle $$ \theta $$, then for any point $$ z $$, the rotated point is given by: $$ R(z) = z e^{i\theta} $$ (c) **Magnification**: A magnification (or dilation) is a function that scales every point in a space by a fixed factor. If $$ M $$ is a magnification by a factor $$ k $$, then for any point $$ z $$, the magnified point is given by: $$ M(z) = k z $$ These definitions cover the requested topics comprehensively. If you have any further questions or need additional details, feel free to ask! Definitions: (a) **Bounded Set**: A set where all points are within a certain distance from each other. (b) **Closed Set**: A set that includes all its limit points. Proof: 3. The function $$ f(z) = z^2 $$ is uniformly continuous in $$ |z| < 1 $$ because for any $$ \epsilon > 0 $$, choosing $$ \delta = \frac{\epsilon}{2} $$ ensures that if $$ |z_1 - z_2| < \delta $$, then $$ |f(z_1) - f(z_2)| < \epsilon $$. 4. The series $$ e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \ldots $$ is absolutely and uniformly convergent because the Weierstrass M-test applies, showing that the series converges uniformly on compact subsets. Maps: (a) **Translation**: Moves every point by a fixed vector. (b) **Rotation**: Rotates every point around a fixed point by a certain angle. (c) **Magnification**: Scales every point by a fixed factor. These definitions and proofs cover the requested topics.

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Calculus Questions