- #1

newgate

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How can I evaluate the following integral please?

##\int_{-\infty}^{+\infty}\frac{e^{i A x}}{\sqrt{x^2+a^2}}dx##

Thank you

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- Thread starter newgate
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In summary, the conversation discusses evaluating an integral using Euler's formula and separating it into two integrals. The integral with sin() is determined to be zero while the integral with cos() is found to be 2*K0(a*A), using the modified Bessel function of the second kind. It is also mentioned that the integral can be simplified by substituting a constant for either A or a.

- #1

newgate

- 13

- 0

How can I evaluate the following integral please?

##\int_{-\infty}^{+\infty}\frac{e^{i A x}}{\sqrt{x^2+a^2}}dx##

Thank you

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- #2

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newgate said:

How can I evaluate the following integral please?

##\int_{-\infty}^{+\infty}\frac{e^{i A x}}{\sqrt{x^2+a^2}}dx##

Thank you

Use Euler to expand the complex exponential into cos() + i*sin().

Separate into two separate integrals. The integrand with sin() is odd, so its integral is ZERO.

Wolfram Alpha won't work the cos() integrand with a and A as symbols, but with a bit of plugging in numbers, one can quickly determine that the integral of the cos() integrand is 2*K0(a*A), where K0 is the modified Bessel function of the second kind. A good integral table would likely tell you that also.

Mathematica or Wolfram Alpha Pro would probably do the original integral.

- #3

newgate

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Thank you Dr.CourtneyDr. Courtney said:Use Euler to expand the complex exponential into cos() + i*sin().

Separate into two separate integrals. The integrand with sin() is odd, so its integral is ZERO.

Wolfram Alpha won't work the cos() integrand with a and A as symbols, but with a bit of plugging in numbers, one can quickly determine that the integral of the cos() integrand is 2*K0(a*A), where K0 is the modified Bessel function of the second kind. A good integral table would likely tell you that also.

Mathematica or Wolfram Alpha Pro would probably do the original integral.

- #4

mathman

Science Advisor

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I am not familiar with Wolfram. However, it is easy enough to modify the integral to get rid of A or a, so that you have only one constant.newgate said:Thank you Dr.Courtney

- #5

newgate

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How? I'm curious :Dmathman said:I am not familiar with Wolfram. However, it is easy enough to modify the integral to get rid of A or a, so that you have only one constant.

- #6

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newgate said:How? I'm curious :D

Substitution ##u = Ax##. Then you just have to find an integral of the type ##\int_{-\infty}^{+\infty} \frac{e^{ix}}{\sqrt{x^2 + C}}dx.##. Likewise, a good substitution will leave you with an integral of the type ##\int_{-\infty}^{+\infty} \frac{e^{Cix}}{\sqrt{x^2 + 1}}dx.##

- #7

newgate

- 13

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Ok thank you very much.micromass said:Substitution ##u = Ax##. Then you just have to find an integral of the type ##\int_{-\infty}^{+\infty} \frac{e^{ix}}{\sqrt{x^2 + C}}dx.##. Likewise, a good substitution will leave you with an integral of the type ##\int_{-\infty}^{+\infty} \frac{e^{Cix}}{\sqrt{x^2 + 1}}dx.##

The integral of exponential over square root is a mathematical operation used to find the area under the curve of a function that contains an exponential term and a square root term. It is typically denoted as ∫[e^x/√(x)]dx.

The integral of exponential over square root is important because it appears in many real-world applications, such as in physics, engineering, and finance. It is also used in solving differential equations and in finding the probability of certain events in statistics.

The integral of exponential over square root can be solved using various techniques, such as integration by substitution, integration by parts, or using special integration formulas. It is important to remember to check for convergence and to use appropriate limits of integration.

The integral of exponential over square root has many applications, including calculating the work done in a system with variable forces, determining the electric potential of a charged particle, and finding the expected value of a continuous random variable in statistics.

Yes, there are some special cases for the integral of exponential over square root. One example is when the limits of integration are from 0 to infinity, in which case the integral can be solved using the Laplace transform. Another special case is when the exponential term is replaced with a trigonometric function, such as in the integral of sine over square root.

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