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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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In summary: Because the area under a complex curve is not simply the product of the areas under the individual curves.

- #1

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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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https://www.physicsforums.com/threads/a-physical-meaning-to-complex-integration.268983/

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THE HARLEQUIN said:

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

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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 ?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

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 )

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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

As I said, the complex plane is a generalization of the real line. Complex functions, though, have

Integration is one of the most important tools in complex analysis and the basis for several important theorems.

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THE HARLEQUIN said:

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.

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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.

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but why doesn't it make sense when the function is complex valued ?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.

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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

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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.

The main difference between real number calculus and complex calculus is that real number calculus deals with functions and variables that have real values, while complex calculus deals with functions and variables that have complex values. This means that complex calculus involves a greater level of mathematical complexity and abstraction compared to real number calculus.

Complex calculus is important in science and mathematics because it allows for the modeling and analysis of complex systems and phenomena. Many physical and natural phenomena, such as electromagnetic fields and fluid dynamics, can be described using complex numbers and require the use of complex calculus for accurate analysis and prediction.

It depends on the specific field of science and research. While some scientists may not need to use complex calculus extensively, others may require a deep understanding of it for their work. For example, physicists and engineers often use complex calculus in their research and experiments.

No, real number calculus and complex calculus are not interchangeable. While both involve the study of calculus principles, they deal with different types of numbers and have different applications. Attempting to use one in place of the other can result in incorrect or nonsensical solutions.

Some common applications of complex calculus include analyzing and predicting the behavior of electrical circuits, solving differential equations in physics and engineering, and studying the properties of fluids and gases in chemistry and biology. It is also used in fields such as signal processing, optics, and quantum mechanics.

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