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.

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!