Logarithmic divergence of an integral

In summary, the integral given is logarithmically divergent due to the higher power of ##k## in the denominator (8) compared to the numerator (4). A more formal proof can be obtained by breaking the polynomial denominator into partial fractions, which will result in a logarithmic term upon integration.
  • #1
spaghetti3451
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I would like to prove that the following integral is logarithmically divergent.

$$\int d^{4}k \frac{k^{4}}{(k^{2}-a)((k-b)^{2}-x)((k-y)^{2}-a)((k-z)^{2}-a)}$$

This is 'obvious' because the power of ##k## in the numerator is ##4##, but the highest power of ##k## in the denominator is ##8##.

However, it is the highest power of ##k## in the denominator that is ##8##. There are other terms in ##k## in the denominator of the form ##k^7##, ##k^6##, etc.

I was wanting a more formal proof that the integral is logarithmically divergent.
 
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  • #2
How does this sound?

Your denominator is a polynomial in k; so it can be written a product of terms like (k - zi) where the zi are zeros of the polynomial. If you break this product into partial fractions, you'll automatically get a log on integration.
 

1. What is logarithmic divergence of an integral?

Logarithmic divergence of an integral is a type of divergence that occurs in certain mathematical integrals when the limit of the integral approaches infinity. It is characterized by a logarithmic function in the denominator of the integrand.

2. How is logarithmic divergence different from other types of divergence?

Logarithmic divergence is different from other types of divergence, such as polynomial or exponential divergence, because it is characterized by a logarithmic function in the integrand. This results in a slower rate of growth compared to other types of divergence.

3. What causes logarithmic divergence?

Logarithmic divergence is caused by a logarithmic function in the integrand of the integral. This can happen when the integrand contains a singularity or when the integrand is proportional to a logarithmic function.

4. How is logarithmic divergence related to the concept of convergence?

Logarithmic divergence is considered a type of divergence, which is the opposite of convergence. When the limit of an integral approaches infinity and the integrand contains a logarithmic function, the integral is said to have logarithmic divergence and does not converge.

5. Are there any applications of logarithmic divergence in science or engineering?

Yes, logarithmic divergence is commonly seen in various fields of science and engineering, such as physics, chemistry, and electrical engineering. It is often used to model and analyze physical phenomena, such as the behavior of particles in a magnetic field or the growth of bacterial populations.

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