Proof of Lemma 2.1, Part (vi) in Palka's Ch.4: Explaining Inequality

In summary, in Chapter 4 of Bruce P. Palka's book "An Introduction to Complex Function Theory", Section 2.2 discusses the properties of contour integrals. In particular, the proof of Lemma 2.1, part (vi), shows that $\int_a^b \text{ Re} \{ u f[ \gamma (t) ] \dot{ \gamma } (t) \} \ dt \le \int_a^b \lvert u f[ \gamma (t) ] \dot{ \gamma } (t) \rvert \ dt$ by using the fact that $\operatorname{Re}(z) \le \lvert z\rvert$ for all $
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I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ...

I am focused on Chapter 4: Complex Integration, Section 2.2 Properties of Contour Integrals ...

I need some further help with some other aspects of the proof of Lemma 2.1, part (vi), Section 2.2, Chapter 4 ...

Lemma 2.1, Chapter 4 reads as follows:View attachment 7433
View attachment 7434In the above text from Palka, near the end of the proof of part (vi), we read the following:" ... ... \(\displaystyle \int_a^b \text{ Re} \{ u f[ \gamma (t) ] \dot{ \gamma } (t) \} \ dt\) \(\displaystyle \le \int_a^b \lvert u f[ \gamma (t) ] \dot{ \gamma } (t) \rvert \ dt\) ... ... "
Can some please explain why/how we have \(\displaystyle \int_a^b \text{ Re} \{ u f[ \gamma (t) ] \dot{ \gamma } (t) \} \ dt \) \(\displaystyle \le \int_a^b \lvert u f[ \gamma (t) ] \dot{ \gamma } (t) \rvert \ d\)t ... ... ?
Help will be much appreciated ...

Peter
 
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It follows from the fact that $\operatorname{Re}(z) \le \lvert z\rvert$ for all $z$.
 
  • #3
Euge said:
It follows from the fact that $\operatorname{Re}(z) \le \lvert z\rvert$ for all $z$.
Oh ... should have seen that ...

... thanks Euge ...

Peter
 

Related to Proof of Lemma 2.1, Part (vi) in Palka's Ch.4: Explaining Inequality

What is Lemma 2.1 in Palka's Chapter 4?

Lemma 2.1 in Palka's Chapter 4 is a mathematical statement that is used to prove a larger theorem or proposition. It is a building block for more complex proofs and is typically a simpler, well-known result that can be applied in a specific context.

What is the significance of Part (vi) in Lemma 2.1?

Part (vi) in Lemma 2.1 is a specific part of the lemma that is being used to prove a larger theorem. It may contain a specific condition or relationship between variables that is crucial for the overall proof.

Why is it important to explain inequality in mathematical proofs?

Inequality is an important concept in mathematics and is often used in proofs to show that one quantity is greater than or less than another. Explaining inequality in a proof helps to clarify the reasoning behind the steps and ensures that the proof is valid.

How can one prove Part (vi) in Lemma 2.1 in Palka's Chapter 4?

To prove Part (vi) in Lemma 2.1, one must carefully follow the logical steps outlined in the proof and use the definitions and properties of the mathematical objects involved. It may also be helpful to use other proven results or theorems to support the proof.

Are there any real-world applications of Lemma 2.1 in Palka's Chapter 4?

It is possible that Lemma 2.1 in Palka's Chapter 4 has real-world applications in various fields such as physics, economics, or engineering. However, the specific context and variables of the lemma may need to be adapted to fit the real-world scenario.

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