Study With Nitin

Quiz Quiz: Find the correct value of * If the number 48327*8 is exactly divisible by 11, then the value of * is: 2 3 9 1 Submit

Complex Analysis Questions Complex Analysis Q. Represent the complex number z = 1 + i√3 in the polar form. Sol. 1 + i√3 = r(cos θ + isinθ) ∴ rcosθ = 1 ...(1) and rsinθ = √3 ...(2) Squring and adding (1) and (2), we get, r 2 (cos 2 θ + sin 2 θ) = 1 + 3 r 2 = 4 ⇒ r = 2 ∴ form (1) and (2), cosθ = 1/2, sinθ = √3/2 ∴ θ = π/3 ∴ z = 1 + i√3 = 2(cos π/3 + isin π/3) The complex number z= 1 + √3 is represented by P(2,π/3) into polar.

Triangle Inequality in Complex Analysis What is the Triangle Inequality? In Complex Analysis, it states that any two complex number z 1 and z 2 , the equality holds: |z 1 + z 2 | ≤ |z 1 | + |z 2 | Here, |z| denotes the modulus of the complex number, and the modulus of complex number z = x + y i , where x is the real number and y i is the imaginary number. So, in complex analysis is defined as, |z| = √(x 2 + y 2 ) What is the Triangle Inequality number theory? In Number Theory, the Triangle Inequality is discussed in the context of the absolute value function over the integers or the real number. It states that for any real number a and b, the Inequality holds: |a + b| ≤ |a| + |b| Proof of Triangle Inequality in Number Theory. Wha...

Triangle Inequality in Number Theory Triangle Inequality in Number Theory For any real numbers a and b , the Triangle Inequality states: |a + b| ≤ |a| + |b| Proof We will consider different cases based on the signs of a and b : Case 1: a ≥ 0 and b ≥ 0 |a + b| = a + b = |a| + |b| Case 2: a ≥ 0 and b < 0 (or vice versa) If a + b ≥ 0, then: |a + b| = a + b ≤ a - b = |a| + |b| If a + b < 0, then: |a + b| = -(a + b) = -a - b = |a| + |b| Case 3: a < 0 and b < 0 |a + b| = -(a + b) = -a - b = |a| + |b| Algebraic Proof Using the definition of absolute values: |a + b| 2 ≤ (|a| + |b|) 2 |a + b| 2 = (a + b) 2 = a 2 + 2ab + b 2 (|a| + |b|) 2 = |a| 2 + 2|a||b| + |b| 2 Since |a| 2 = a 2 and |b| 2 = b 2 : a 2 + 2ab + b 2 ≤ a 2 + 2|a||b| + b 2 ...

Number System Number System We know that ten digits are used to write a number. There numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. To write a number, we take units like: tens, hundreds, thousand and etc. form right to left respectively. Example 1: Write the following numbers in words: 555555 Five hundred fifty-five thousand five hundred fifty-five. 978779 Nine hundred seventy-eight thousand seven hundred seventy-nine. 9787987 Nine million seven hundred eighty-seven thousand nine hundred eighty-seven. Example 2: Write the following numbers in words: ...

Cauchy-Riemann equations Proof of Cauchy-Riemann Equation

Euler's Theorem of Homogeneous Functions Euler's Theorem of Homogeneous Functions Euler's theory of equivalent functions clarifies a key concept in statistical analysis, showing the relationship between equivalent functions. Discover deeper insights into the scaling behavior of these functions and their impact on mathematical models, optimization, and applications. Unpack the basic principles of Euler's theorem and dive clearly and deeply into the fascinating world of identical functions. If \(z\) be a homogeneous function of \(x, y\) of order \(n\), then \(x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=nz\) all Proof: Since \(z\) is a homogeneous function of \(x, y\) of order \(n\). \(z=x^{n}f(\frac{y}{x}) \dots (1)\) \(\frac{\partial z}{\partial x}=n~x^{n-1}f(\frac{y}{x})+x^{n}.f^{\prime}(\frac{y}{x}).(-\frac{y}{x^{2}}).\) or ...