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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 help with some 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 7429
View attachment 7430In the above text from Palka, at the start of the proof of (vi), we read the following:
" ... ... We suppose that $$\int_{ \gamma } f(z)dz \ne 0$$ - (vi) holds trivially otherwise - and set $$u = e^{ -i \theta }$$ , where $$\theta$$ is any argument of $$\int_{ \gamma } f(z)dz$$. Thus $$\lvert u \rvert = 1$$ and $$\left\lvert \int_{ \gamma } f(z)dz \right\rvert = u \int_{ \gamma } f(z)dz$$ ... ... "
My questions are as follows:
Question 1
Is $$u = e^{ -i \theta }$$ a simple change of variable process?
Question 2
What is meant by " ... $$\theta$$ is any argument of $$\int_{ \gamma } f(z)dz$$" ... ?
Question 3
Can someone please explain why/how $$\left\lvert \int_{ \gamma } f(z)dz \right\rvert = u \int_{ \gamma } f(z)dz$$ ... ...?
Help will be much appreciated ..
Peter
I am focused on Chapter 4: Complex Integration, Section 2.2 Properties of Contour Integrals ...
I need help with some 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 7429
View attachment 7430In the above text from Palka, at the start of the proof of (vi), we read the following:
" ... ... We suppose that $$\int_{ \gamma } f(z)dz \ne 0$$ - (vi) holds trivially otherwise - and set $$u = e^{ -i \theta }$$ , where $$\theta$$ is any argument of $$\int_{ \gamma } f(z)dz$$. Thus $$\lvert u \rvert = 1$$ and $$\left\lvert \int_{ \gamma } f(z)dz \right\rvert = u \int_{ \gamma } f(z)dz$$ ... ... "
My questions are as follows:
Question 1
Is $$u = e^{ -i \theta }$$ a simple change of variable process?
Question 2
What is meant by " ... $$\theta$$ is any argument of $$\int_{ \gamma } f(z)dz$$" ... ?
Question 3
Can someone please explain why/how $$\left\lvert \int_{ \gamma } f(z)dz \right\rvert = u \int_{ \gamma } f(z)dz$$ ... ...?
Help will be much appreciated ..
Peter
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