Question on Chapter 7 : Penrose Road to Reality

[smodak]

A little about my background. I have a undergraduate degree in physics which I obtained exactly 20 years back. I also have a undergraduate degree in EE which I obtained about 17 years back. I have worked as a software engineer all my life and currently also pursuing my MBA in Finance. Needless to say that I am a bit rusty in Physics and math but I have always been interested in Physics and recently started reading Physics again after I read (and enjoyed) the theoretical minimum book by Susskind.

I started reading The Road to Reality and so far like it very much. In introducing the contour integration on a complex plane, he mentions a few things that I do not understand.

1. In Chapter 7, page 124 he says (citing a definite integral of a real function between a and b)

There is only one way to go from a to b along the real line

But for a complex plane

We do not just have one route from a to b

Question: Why do we have only one path in a real plane but more than one on a complex plane?

2. He says that he will introduce C-R equations in chapter 12 but he then says that the C-R equations tell us that

If we do our integration along one such path then we get the same answer as along any other such path that can be obtained from the first by continuous deformation within the domain of the function.

Question: I do not understand this statement

Could you help? Please be gentle :)

Is there a book on complex analysis (not too thick) that will help me understand Chapter 7 (or later chapters) in Penrose?

 

[Dale]

Question: Why do we have only one path in a real plane but more than one on a complex plane?

There is only one path on the real LINE. There are multiple paths on the complex PLANE.

 

[lightarrow]

I started reading The Road to Reality and so far like it very much. In introducing the contour integration on a complex plane, he mentions a few things that I do not understand.

1. In Chapter 7, page 124 he says (citing a definite integral of a real function between a and b)
<<There is only one way to go from a to b along the real line>>

But for a complex plane
<<We do not just have one route from a to b>>

Question: Why do we have only one path in a real plane but more than one on a complex plane?

Because to go from the real number a to the real number b integrating in the real line you can only go from a to a +dx, then a+2dx...to b.
In the complex plane, being ... a plane, you can go also "up and down" the real line, to the second dimension of the imaginary numbers, so you can follow even a "complex" (not a casually chosen term ) contour from the point (a,0) to the point (b,0) of the plane.

2. He says that he will introduce C-R equations in chapter 12 but he then says that the C-R equations tell us that
<<If we do our integration along one such path then we get the same answer as along any other such path that can be obtained from the first by continuous deformation within the domain of the function.>>

It's true for an "analythic" function, not for all complex functions: if in the interval (a,b) there is a point in which the function is not defined or goes to infinite, the theorem (see down) is not valid.

Question: I do not understand this statement

See "Cauchy theorem".
P.S.
You, as most of people, even with degrees in physics and mathematics, will be able to read that book just up to a certain point, then ... oblivion

--
lightarrow

 

[WannabeNewton]

Question: I do not understand this statement

He is just stating, in loose terms, the homotopy invariance of the line integral. See here: http://math.ucr.edu/~res/math246A-2012/integrals2012.pdf

 

[smodak]

Thanks guys. That was very helpful!