Real number calculus vs complex calculus

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Discussion Overview

The discussion revolves around the conceptual differences between real number calculus and complex calculus, particularly focusing on complex integration. Participants explore the meaning of integration in both contexts, the analogy between area under curves in real integrals and complex integrals, and the implications of complex functions' properties.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants express confusion about the analogy between area under curves in real integrals and complex integration, noting that the concept of area does not directly translate to the complex plane.
  • Others suggest that complex integration can be viewed as a variant of path integration, which may have interpretations related to area under curves in certain contexts.
  • One participant points out that while the complex plane can be viewed as a two-dimensional Cartesian plane, the properties of complex functions differ significantly from real functions, particularly regarding differentiability and integration.
  • It is noted that complex functions must satisfy stricter conditions for differentiability, leading to the conclusion that integrals of analytic functions are path-independent.
  • Some participants argue that for real-valued functions, one can consider the volume under a surface, but for complex-valued functions, this concept becomes less meaningful.
  • There is a discussion about the nature of complex line integrals, which can be interpreted as combining two regular integrals, emphasizing the need to think in terms of path integrals rather than area.
  • One participant highlights the significance of the residue theorem and winding numbers in complex integration, suggesting that these concepts are central to understanding integrals of analytic functions.

Areas of Agreement / Disagreement

Participants express a range of views on the relationship between real and complex integration, with no clear consensus on how to best conceptualize complex integration in relation to area. Some agree on the importance of path integrals, while others remain uncertain about the implications of complex-valued functions.

Contextual Notes

Participants mention various assumptions about the nature of complex functions and integration, including the distinction between real and complex values, the implications of differentiability, and the interpretation of integrals in different contexts. These assumptions may affect the understanding of the discussion.

THE HARLEQUIN
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I have started studying complex integration recently and i just can't seem to get the things in my head .
the biggest problem i am facing is that :
when solving real number integrals the area under the curve of the function is what integration means ...
but i can't seem to find an analogy between this and complex integration ...
 
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THE HARLEQUIN said:
I have started studying complex integration recently and i just can't seem to get the things in my head .
the biggest problem i am facing is that :
when solving real number integrals the area under the curve of the function is what integration means ...
but i can't seem to find an analogy between this and complex integration ...

Consult the book "visual complex analysis" by Needham.

Also, it is a variant of path integration, so you should first be comfortable with that. Path integrals can be easily interpreted as an area under a curve. http://en.wikipedia.org/wiki/Line_integral#mediaviewer/File:Line_integral_of_scalar_field.gif
 
micromass said:
Also, it is a variant of path integration, so you should first be comfortable with that. Path integrals can be easily interpreted as an area under a curve. http://en.wikipedia.org/wiki/Line_integral#mediaviewer/File:Line_integral_of_scalar_field.gif
i thought complex plane is a 2 dimensional plane just like our cartesian plane ... so why can't i just take the area under the curve in complex plane here too ?
and while approaching non complex integration why don't we integrate for any two points on the plane with an arbitrary path ? ( sorry if i am wrong about everything i said , i am still a noob at complex integration )
 
THE HARLEQUIN said:
i thought complex plane is a 2 dimensional plane just like our cartesian plane ... so why can't i just take the area under the curve in complex plane here too ?
and while approaching non complex integration why don't we integrate for any two points on the plane with an arbitrary path ? ( sorry if i am wrong about everything i said , i am still a noob at complex integration )

Yes and no. Yes it is a 2-dimensional cartesian plane when viewed a certain way. No because it is a generalization of the real line when viewed another way.

Example: In the 2-dimensional cartesian plane you can define addition and subtraction of points easily, but there is no straightforward way of defining a way to multiply two points and get a new point. In the complex plane, multiplication is defined from the beginning.

As I said, the complex plane is a generalization of the real line. Complex functions, though, have stricter requirements than real functions. For a real function to be differentiable, the right- and left-hand derivative must both exist and be equal. For a complex function, the derivatives must be equal no matter how we approach the point in question. Such functions are called analytic, and they have several interesting properties, one of them being that the integral from one point to another is not dependent on the path used. This again means that the integral of an analytic function along a closed curve is zero.

Integration is one of the most important tools in complex analysis and the basis for several important theorems.
 
THE HARLEQUIN said:
I have started studying complex integration recently and i just can't seem to get the things in my head .
the biggest problem i am facing is that :
when solving real number integrals the area under the curve of the function is what integration means ...
but i can't seem to find an analogy between this and complex integration ...

If a function in the plane were real valued then one could draw its graph in three dimensions. It would look like a surface and it definitely makes sense to talk about the volume underneath it (not area). But the only real valued continuous complex differentiable functions are constants. So for them computing these volumes is uninteresting.

You should try to convince yourself that complex differentiable functions that are real valued are constants. It is not hard.
 
THE HARLEQUIN said:
i thought complex plane is a 2 dimensional plane just like our cartesian plane ... so why can't i just take the area under the curve in complex plane here too ?
and while approaching non complex integration why don't we integrate for any two points on the plane with an arbitrary path ? ( sorry if i am wrong about everything i said , i am still a noob at complex integration )

There is nothing to stop you from taking the area under a curve if the function is real valued. If it is complex valued the idea makes no sense.
 
Last edited:
lavinia said:
There is nothing to stop you from taking the area under a curve if the function is real valued. If it is complex valued the idea makes no sense.
but why doesn't it make sense when the function is complex valued ?
 
THE HARLEQUIN said:
but why doesn't it make sense when the function is complex valued ?

A complex number is not a height.

A complex line integral can be thought of as two regular integrals added together (after multiplying one of them by i) to get a complex number
 
  • #10
i agree one should think in terms of path integrals, not area. consider the path integral of -ydx/(x^2+y^2) + xdy/(x^2+y^2) around a loop missing the origin. If I remembered the formula correctly, this is just dtheta, so the integral gives you the change in angle of a ray emanating from the origin, as you go around this curve, i.e. 2pi times the winding number. this can be computed in stages as actual angel changes for continuously chosen branches of the angle function theta.

in general, every complex analytic function f is locally of form f = dg for some analytic function g, so integrating f along a path is computed from the changes in value of these locally given g's in stages along this path.

the catchy-residue formula tells you that the integral of a local quotient of analytic functions is always equivalent to just computing winding numbers and residues at points where the denominator equals zero.
 

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