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- Jun 22, 2012

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I am reading the book: Complex Analysis: A First Course with Applications (Third Edition) by Dennis G. Zill and Patrick D. Shanahan ...

I need some help with an aspect of the proof of Theorem 3.1.1 (also named Theorem A1 and proved in Appendix 1) ...

The statement of Theorem 3.1.1 (A1) reads as follows:

In the proof of Theorem 3.1.1 (A1) [see below] we read the following:

" ... ... On the other hand, with the identifications \(\displaystyle f(z) = u(x,y) + i v(x,y)\) and \(\displaystyle L = u_0 + i v_0\), the triangle inequality gives

\(\displaystyle \mid f(z) - L \mid \ \leq \ \mid u(x,y) - u_0 \mid + \mid v(x,y) - v_0 \mid\) ... ... "

My question is as follows:

How exactly do we apply the triangle inequality to get \(\displaystyle \mid f(z) - L \mid \ \leq \ \mid u(x,y) - u_0 \mid + \mid v(x,y) - v_0 \mid\) ... ... ?

Note: Zill and Shanahan give the triangle inequality as

\(\displaystyle \mid z_1 + z_2 \mid \ \leq \ \mid z_1 \mid + \mid z_2 \mid\)

My thoughts are as follows:

\(\displaystyle \mid f(z) - L \mid \ = \ \mid u(x,y) + i v(x,y) - (u_0 + i v_0 ) \mid \ = \ \mid ( u(x,y) - u_0 ) + i ( v(x,y) - v_0 ) \mid\)

so in triangle inequality put

\(\displaystyle z_1 = u(x,y) - u_0 + i.0\)

and

\(\displaystyle z_2 = 0 + i ( v(x,y) - v_0 \))

and apply triangle inequality ...

Is that correct?

Hope someone can help ...

Peter

==================================================================================

The statement and proof of Theorem 3.1.1 (given in Appendix 1 where the theorem is called Theorem A.1) reads as follows:

Hope that helps ...

Peter

I need some help with an aspect of the proof of Theorem 3.1.1 (also named Theorem A1 and proved in Appendix 1) ...

The statement of Theorem 3.1.1 (A1) reads as follows:

In the proof of Theorem 3.1.1 (A1) [see below] we read the following:

" ... ... On the other hand, with the identifications \(\displaystyle f(z) = u(x,y) + i v(x,y)\) and \(\displaystyle L = u_0 + i v_0\), the triangle inequality gives

\(\displaystyle \mid f(z) - L \mid \ \leq \ \mid u(x,y) - u_0 \mid + \mid v(x,y) - v_0 \mid\) ... ... "

My question is as follows:

How exactly do we apply the triangle inequality to get \(\displaystyle \mid f(z) - L \mid \ \leq \ \mid u(x,y) - u_0 \mid + \mid v(x,y) - v_0 \mid\) ... ... ?

Note: Zill and Shanahan give the triangle inequality as

\(\displaystyle \mid z_1 + z_2 \mid \ \leq \ \mid z_1 \mid + \mid z_2 \mid\)

My thoughts are as follows:

\(\displaystyle \mid f(z) - L \mid \ = \ \mid u(x,y) + i v(x,y) - (u_0 + i v_0 ) \mid \ = \ \mid ( u(x,y) - u_0 ) + i ( v(x,y) - v_0 ) \mid\)

so in triangle inequality put

\(\displaystyle z_1 = u(x,y) - u_0 + i.0\)

and

\(\displaystyle z_2 = 0 + i ( v(x,y) - v_0 \))

and apply triangle inequality ...

Is that correct?

Hope someone can help ...

Peter

==================================================================================

The statement and proof of Theorem 3.1.1 (given in Appendix 1 where the theorem is called Theorem A.1) reads as follows:

Hope that helps ...

Peter

Last edited: