Hot Integral - 17

Calculus Level 5

\[\large \int\limits_{-\infty}^{\infty} (f(x) + g(x))e^{-2\pi i \lambda x} dx = \Re - i\Im\]

where

\(\displaystyle f(x)=\frac{1+2x^2(\sin(x^2)+\log(|x|))}{2x}\)

\(\displaystyle g(x)=\frac{sgn^2(x-\frac{1}{2})+sgn^2(x+\frac{1}{2})}{4}-\frac{sgn(x-\frac{1}{2})sgn(x+\frac{1}{2})}{2}\)

\[\displaystyle \Re = \frac{\sin^a(\lambda \pi)}{(\lambda^b\pi^c)} + \frac{d}{e\lambda^f|\lambda|} - \gamma\delta^{(h)}(\lambda)\]

\[\displaystyle \Im = \pi^j(\lambda^k\pi^{\frac{l}{m}}\cos((\lambda\pi)^n - \frac{\pi^p}{q}) + \frac{r}{s}sgn(\frac{\lambda}{t})) \]

Find \(a+b+c+d+e+f+h+j+k+l+m+n+p+q-r+s+t\)

Clarifications:

  • \(\delta^{(n)}(\cdot)\) represents the \(n^{th}\) derivative of delta function.

  • \(i=\sqrt{-1}\)

  • \(|\cdot|\) represents modulus function.

  • \(sgn(x)=\frac{x}{|x|}\)

  • \(\Re , \Im\) means real and imaginary part respectively.

  • \(\gamma\) is Euler -Mascheroni constant.


This is a part of Hot Intergals

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