# 13.1 Illustration of $E_i(y)=\int_0^\infty\delta(x-1)\,Ei(y\,x)\,dx\,,y<0$

This page illustrates a formula for $E_i(y)$ derived from evaluation of the following Mellin convolution using the Fourier series representation of $\delta(x-1)$.

(1) $\quad E_i(y)=\int_0^\infty\delta(x-1)\,E_i(y\,x)\,dx\,,\quad y<0$

The blue and dashed-gray curves in the following plot represent the evaluation of formula (1) for $E_i(y)$ and the reference function $E_i(y)$ respectively. Note formula (1) for $E_i(y)$ only converges for $y<0$.

The red and blue curves in the following  plot illustrate that formula (1) for $E_i(y)$ converges as the evaluation limit increases towards infinity, and the dashed-gray curve illustrates the reference function $E_i(y)$. The evaluation limit of formula (1) for the blue curve is a little over twice the evaluation limit of formula (1) for the red curve, and consequently the blue curve converges to the dashed-gray reference function $E_i(y)$ much better than the red curve.

The orange curve in the following plot represents the real part of formula (1) for $E_i(y)$ evaluated along the line $y=-\frac{1}{2}+i\,t$, and the blue curve represents the corresponding reference function $\Re(E_i(y))=\Re(-E_1(-y))$. Note the real part of formula (1) for $E_i(y)$ converges to the reference function $\Re(E_i(y))=\Re(-E_1(-y))$.

The orange curve in the following plot represents the imaginary part of formula (1) for $E_i(y)$ evaluated along the line $y=-\frac{1}{2}+i\,t$. The blue curve represents the corresponding reference function $\Im(E_i(y))$. Note the imaginary part of formula (1) for $E_i(y)$ does not converge to the reference function $\Im(E_i(y))$ due to the branch point of $\Im(E_i(-\frac{1}{2}+i\,t))$ at $t=0$.

The orange curve in the following plot represents the imaginary part of formula (1) for $E_i(y)$ evaluated along the line $y=-\frac{1}{2}+i\,t$. The blue curve represents the corresponding reference function $\Im(-E_1(-y))$. Note the imaginary part of formula (1) for $E_i(y)$ converges to the reference function $\Im(-E_1(-y))$.