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I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ...
I am currently focussed on Chapter III Elementary Properties and Examples of Analytic Functions ... Section 2: Analytic Functions ... ...
I need help in fully understanding aspects of Theorem 2.29 ...
[NOTE: Notice that the statement of Theorem 2.29 follows the proof ... see below ... ]
Theorem 2,29 and its proof read as follows:
View attachment 7402
https://www.physicsforums.com/attachments/7403
In the above text by Conway, we read the following:
" ... ... Using the fact that $$u$$ and $$v$$ satisfy the Cauchy-Riemann equations it is easy to see that
$$\frac{ f( z + s + it ) - f(z) }{ s + it } = u_x(z) + iv_x(z) + \frac{ \phi(s, t) + i \psi(x, t) }{ s + it }$$ ... ... ... "
My question is as follows:
What is the (rigorous) process by which we arrive at
$$\frac{ f( z + s + it ) - f(z) }{ s + it } = u_x(z) + iv_x(z) + \frac{ \phi(s, t) + i \psi(x, t) }{ s + it }$$ ...?
That is ... why/how exactly does this follow ...
Hope someone can help ... I am aiming to have a rigorous understanding of the above proof ...
Peter***EDIT***I have been reflecting on the issue/problem above ...
Part of the answer may well be as follows:
$$\frac{ f(z + s + it) - f(z) }{ s + it } $$$$= \frac{ f(x + iy + s + it) - f(x + iy) }{ s + it }$$ $$= \frac{ [ u( x + s, y + t ) + iv( x + s, y + t ) ] - [ u(x, y) + iv(x, y) ] }{ s + it }$$
$$= \frac{ [ u( x + s, y + t ) - u(x, y) ] + i [ v( x + s, y + t ) - v(x, y) ] }{ s + it }$$$$= \frac{ [ u_x ( x, y )s + u_y (x, y)t + \phi (s, t) }{ s + it } + i \frac{ [v_x ( x, y )s + v_y (x, y)t + \psi (s, t) }{ s + it }$$$$ = \frac{ [ u_x ( x, y )s - v_x (x, y)t + \phi (s, t) }{ s + it } + i \frac{ [v_x ( x, y )s + u_x (x, y)t + \psi (s, t) }{ s + it }$$ $$= \frac{ [ u_x ( x, y )s + i u_x (x, y)t ] }{ s + it } + i \frac{ [v_x ( x, y )s + i v_x (x, y)t ] }{ s + it } + \frac{ [ \phi (s, t) + i \psi (s, t)] }{ s + it } $$$$= u_x(z) + iv_x(z) + \frac{ [ \phi (s, t) + i \psi (s, t)] }{ s + it }$$Is that correct?
======================================================================================It may help readers of the above post to have access to Conway's introduction to the Cauchy-Riemann conditions (necessity case) ... so I am providing the same ... as follows:
View attachment 7404
View attachment 7405
I am currently focussed on Chapter III Elementary Properties and Examples of Analytic Functions ... Section 2: Analytic Functions ... ...
I need help in fully understanding aspects of Theorem 2.29 ...
[NOTE: Notice that the statement of Theorem 2.29 follows the proof ... see below ... ]
Theorem 2,29 and its proof read as follows:
View attachment 7402
https://www.physicsforums.com/attachments/7403
In the above text by Conway, we read the following:
" ... ... Using the fact that $$u$$ and $$v$$ satisfy the Cauchy-Riemann equations it is easy to see that
$$\frac{ f( z + s + it ) - f(z) }{ s + it } = u_x(z) + iv_x(z) + \frac{ \phi(s, t) + i \psi(x, t) }{ s + it }$$ ... ... ... "
My question is as follows:
What is the (rigorous) process by which we arrive at
$$\frac{ f( z + s + it ) - f(z) }{ s + it } = u_x(z) + iv_x(z) + \frac{ \phi(s, t) + i \psi(x, t) }{ s + it }$$ ...?
That is ... why/how exactly does this follow ...
Hope someone can help ... I am aiming to have a rigorous understanding of the above proof ...
Peter***EDIT***I have been reflecting on the issue/problem above ...
Part of the answer may well be as follows:
$$\frac{ f(z + s + it) - f(z) }{ s + it } $$$$= \frac{ f(x + iy + s + it) - f(x + iy) }{ s + it }$$ $$= \frac{ [ u( x + s, y + t ) + iv( x + s, y + t ) ] - [ u(x, y) + iv(x, y) ] }{ s + it }$$
$$= \frac{ [ u( x + s, y + t ) - u(x, y) ] + i [ v( x + s, y + t ) - v(x, y) ] }{ s + it }$$$$= \frac{ [ u_x ( x, y )s + u_y (x, y)t + \phi (s, t) }{ s + it } + i \frac{ [v_x ( x, y )s + v_y (x, y)t + \psi (s, t) }{ s + it }$$$$ = \frac{ [ u_x ( x, y )s - v_x (x, y)t + \phi (s, t) }{ s + it } + i \frac{ [v_x ( x, y )s + u_x (x, y)t + \psi (s, t) }{ s + it }$$ $$= \frac{ [ u_x ( x, y )s + i u_x (x, y)t ] }{ s + it } + i \frac{ [v_x ( x, y )s + i v_x (x, y)t ] }{ s + it } + \frac{ [ \phi (s, t) + i \psi (s, t)] }{ s + it } $$$$= u_x(z) + iv_x(z) + \frac{ [ \phi (s, t) + i \psi (s, t)] }{ s + it }$$Is that correct?
======================================================================================It may help readers of the above post to have access to Conway's introduction to the Cauchy-Riemann conditions (necessity case) ... so I am providing the same ... as follows:
View attachment 7404
View attachment 7405
Last edited: