Evaluating a 'WolframAlphan' integral

Question

This problem has to be the real and integral part of Christmas... :)

Here goes:

What is the minimum value of $k$ such that $$\Re\left\{\int_{e^{\large 1/\pi}}^{\pi^{\large 1/e}}\left[% \left(\frac{x^e}{\pi^{x}}\right)^{ex/\pi} - \left(\frac{e^{x}}{x^{\pi}}\right)^{\pi x/e}\right]^{k}\,\mathrm{d}x\right\} = 1? $$

edit: @DonnyFrank has provided a visualization showing that $k$ can take infinitely many values

My Observations

> The integral is negative for odd integers of $k$ and positive for even integers of $k$.

> The integral is complex for non-integer $k$ (it is already complex for any value of $k$ ;)

> The real part of the integral is very close to $1$ as $k$ approaches $10$ - in fact it surpasses $10$ at around $k=9.897$.

My Attempt

No clue. I know that $k$ is not an integer so the binomial theorem wouldn't help at all.

Merry Christmas and a Happy New Year!

Answer 1

Here's a plot of your function:

As you can see there are infinite values of k such that f(k)=1, the first of which is ~9.89577716596896.

Code used to generate this plot:

from mpmath import *
import matplotlib.pyplot as plt
import numpy as np

def f(k):
    return re(quad(lambda x: ((x**e / pi**x)**(e*x/pi) - (e**x / x**pi)**(pi*x/e))**k, [e**(1/pi), pi**(1/e)]))

kl = np.linspace(0,20,1000)
yl = [f(k) for k in kl]

plt.plot(kl, yl, label='f(k)')
plt.plot((0, 20), (1, 1), label='f(k)=1')
plt.legend()
plt.xlabel('k')
plt.ylabel('f(k)')

plt.show()

Finding the first value such that f(k)=1:

print(findroot(lambda k: f(k) - 1, 9.0))

I doubt any "nice" representation can be found for this value, unless you have a particular reason one should exist.