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(...
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What is the unit digit in the number of (2137)753?

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Exam Questions What is the unit digit in the number of (2137) 753 ? a) 1 b) 2 c) 7 d) 9 Answer: To find the unit dight in the number(2137) 753 , we need to focus on the circle of unit digits in the number are raised to powers. Here, we have to focus on the when the power rised and which number is in the unit place. The unit digit of 2137 is 7. When the number 7 will rise the power as following patterns are repeats. 7 1 has a unit digit of 7 7 2 has a unit digit of 9 7 3 has a unit digit of 3 7 4 has a unit digit of 1 7 5 has a unit digit of 7 SInce, the pattern repeats every four powers, now we can find the reminder when 753 is divided by 4: 753/4 = 188 and with give a reminder of 1. This means that the unit digit of (2137) 753 will be same as unit dight of (2137) 1 ∴ the unit digit of (2137) 753 . That is option c is correct.
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