ϕ(t)=∫ −∞∞​ e itx a 2 +(x−it) 2 An​ dx

To solve the given integral, we can start by simplifying the expression inside the integral. Let's expand the denominator term (a^2 + (x - it)^2) using the formula for the sum of squares:

a^2 + (x - it)^2 = a^2 + x^2 - 2ixt + t^2

Now, we can rewrite the integral as:

ϕ(t) = ∫ -∞∞ e^(itx) / (a^2 + x^2 - 2ixt + t^2) dx

Next, we can complete the square in the denominator by adding and subtracting (xt)^2:

ϕ(t) = ∫ -∞∞ e^(itx) / [(x^2 - 2ixt + t^2) + a^2 - (xt)^2] dx

Using the identity (a + b)(a - b) = a^2 - b^2, we can rewrite the denominator as:

ϕ(t) = ∫ -∞∞ e^(itx) / [(x - it + xt)(x + it - xt) + a^2] dx

Now, we can use partial fraction decomposition to split the fraction into simpler terms. Let's assume that the integral can be written as:

ϕ(t) = A ∫ e^(itx) / (x - it + xt) dx + B ∫ e^(itx) / (x + it - xt) dx

To find the values of A and B, we can multiply both sides of the equation by the denominators and then equate the coefficients of the terms involving e^(itx). This will give us a system of equations that we can solve for A and B.

Once we have determined the values of A and B, we can integrate each term separately. The integral of e^(itx) / (x - it + xt) can be evaluated using techniques such as integration by parts or contour integration, depending on the specific values of a, x, and t.

Finally, we can substitute the values of A and B back into the equation and simplify the expression to obtain the final result for ϕ(t).