Problems with identity in complex calc

In summary, the conversation discusses the identity \int_{-\infty}^{\infty} d x \sqrt{(x-i\epsilon)^2-1}= I_+ - I_- + i \int_{-1}^{1}(\ldots), where I_+ = \int_1^{\infty}(\ldots), I_=\int_{-\infty}^{-1}(\ldots), and \epsilon is a small positive number. The problem at hand is to determine why there is a discrepancy in the sign between I_+ and I_-. The author believes the integrand for both should be \sqrt{x^2-1}, but further calculations show that it is definitely real.
  • #1
betel
318
0
Hello,

in a paper I have the identity

[tex] \int_{-\infty}^{\infty} d x \sqrt{(x-i\epsilon)^2-1}= I_+ - I_- + i \int_{-1}^{1}(\ldots) [/tex]


where [tex]I_+ = \int_1^{\infty}(\ldots), I_=\int_{-\infty}^{-1}(\ldots)[/tex] and [tex] \epsilon[/tex] is a small positive number that will be taken to zero at the end.

My Problem is to get the minus sign between [tex]I_+[/tex] and [tex]I_-[/tex]. In all my calculations I get +. The integrand for both should be [tex]\sqrt{x^2-1}[/tex].

Can anybody tell me what I am missing here to get the correct sign.
Thanks in advance.
betel
 
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  • #2
I think it's essential what you (or the author) means by "(...)" for the integrand.
 
  • #3
In further calculations the author uses [tex] \sqrt{x^2-1}[/tex]. The integrand is definitely real.
 

Related to Problems with identity in complex calc

1. What are the main challenges in understanding complex calc identity problems?

The main challenge in understanding complex calc identity problems is the highly abstract and non-intuitive nature of the subject. Complex calc deals with complex numbers, which have both real and imaginary components, and it can be difficult to visualize or conceptualize these numbers. Additionally, the rules and properties of complex numbers are quite different from those of real numbers, making it challenging to apply familiar mathematical concepts.

2. How do identity problems in complex calc differ from those in other areas of math?

Identity problems in complex calc are unique in that they involve complex numbers and their properties. In other areas of math, such as algebra or geometry, identity problems may involve solving equations or proving geometric relationships. In complex calc, identity problems often involve manipulating complex numbers and verifying their properties through algebraic or geometric methods.

3. What are some common mistakes people make when dealing with complex calc identity problems?

One common mistake is assuming that the properties of real numbers also apply to complex numbers. For example, in real numbers, the commutative property of addition states that a + b = b + a, but in complex numbers, this may not always be true. Another mistake is not fully understanding the properties of complex numbers, such as the fact that the product of two complex conjugates is always a real number.

4. How can a better understanding of identity problems in complex calc benefit other areas of math?

A better understanding of identity problems in complex calc can benefit other areas of math by providing a deeper understanding of the relationships between real and complex numbers. This can lead to new insights and approaches in solving problems in other areas of math, such as algebra and geometry. Additionally, complex calc is often used in fields such as physics and engineering, so a strong understanding of its concepts and properties can be beneficial in these fields as well.

5. What resources are available for learning about identity problems in complex calc?

There are various online resources, textbooks, and courses available for learning about identity problems in complex calc. Many universities offer courses on complex analysis, which covers the fundamentals of complex calc. Additionally, there are numerous online tutorials and videos that explain complex calc concepts and problem-solving strategies. It can also be helpful to practice solving problems and seeking guidance from a math tutor or professor if needed.

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