<p>Illustration of bracketing
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\exp(-x)}{2}}\!~\ln \!\left(1+{\frac {2}{x}}\right)&lt;{\rm {E}}_{1}(x)&lt;\exp(-x)\!~\ln \!\left(1+{\frac {1}{x}}\right)~~~,~~~~~x\!&gt;\!0}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mi>exp</mi><mo>⁡<!-- ⁡ --></mo><mo stretchy="false">(</mo><mo>−<!-- − --></mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></mrow><mspace width="negativethinmathspace"></mspace><mtext> </mtext><mi>ln</mi><mspace width="negativethinmathspace"></mspace><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>2</mn><mi>x</mi></mfrac></mrow></mrow><mo>)</mo></mrow><mo>&lt;</mo><msub><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">E</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mi>exp</mi><mo>⁡<!-- ⁡ --></mo><mo stretchy="false">(</mo><mo>−<!-- − --></mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="negativethinmathspace"></mspace><mtext> </mtext><mi>ln</mi><mspace width="negativethinmathspace"></mspace><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>,</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>x</mi><mspace width="negativethinmathspace"></mspace><mo>&gt;</mo><mspace width="negativethinmathspace"></mspace><mn>0</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\frac {\exp(-x)}{2}}\!~\ln \!\left(1+{\frac {2}{x}}\right)&lt;{\rm {E}}_{1}(x)&lt;\exp(-x)\!~\ln \!\left(1+{\frac {1}{x}}\right)~~~,~~~~~x\!&gt;\!0}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1501baf2e30b51995239e4f67c17180ee8ce24b5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:64.91ex; height:6.343ex;" alt="{\displaystyle {\frac {\exp(-x)}{2}}\!~\ln \!\left(1+{\frac {2}{x}}\right)&lt;{\rm {E}}_{1}(x)&lt;\exp(-x)\!~\ln \!\left(1+{\frac {1}{x}}\right)~~~,~~~~~x\!&gt;\!0}"></span></dd></dl><p>of the <a href="https://en.wikipedia.org/wiki/exponential_integral" class="extiw" title="w:exponential integral">exponential integral</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~{\rm {E}}_{1}(x)~}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mtext> </mtext><msub><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">E</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext> </mtext></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle ~{\rm {E}}_{1}(x)~}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/374219449e43743a65c827600f46a9396424d877" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.937ex; height:2.843ex;" alt="{\displaystyle ~{\rm {E}}_{1}(x)~}"></span> (black) with elementary functions 
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\exp(-x)}{2}}\!~\ln \!\left(1+{\frac {2}{x}}\right)~}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mi>exp</mi><mo>⁡<!-- ⁡ --></mo><mo stretchy="false">(</mo><mo>−<!-- − --></mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></mrow><mspace width="negativethinmathspace"></mspace><mtext> </mtext><mi>ln</mi><mspace width="negativethinmathspace"></mspace><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>2</mn><mi>x</mi></mfrac></mrow></mrow><mo>)</mo></mrow><mtext> </mtext></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\frac {\exp(-x)}{2}}\!~\ln \!\left(1+{\frac {2}{x}}\right)~}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b83050c1144fa7b804ee58042331c918ec4afa41" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:22.413ex; height:6.343ex;" alt="{\displaystyle {\frac {\exp(-x)}{2}}\!~\ln \!\left(1+{\frac {2}{x}}\right)~}"></span> (blue)
and
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\exp(-x)\!~\ln \!\left(1+{\frac {1}{x}}\right)~}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mtext> </mtext><mi>exp</mi><mo>⁡<!-- ⁡ --></mo><mo stretchy="false">(</mo><mo>−<!-- − --></mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="negativethinmathspace"></mspace><mtext> </mtext><mi>ln</mi><mspace width="negativethinmathspace"></mspace><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mrow><mo>)</mo></mrow><mtext> </mtext></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle ~\exp(-x)\!~\ln \!\left(1+{\frac {1}{x}}\right)~}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02bde4dea2fde35e3c2614c18a9e68126b7e1746" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:22.545ex; height:6.176ex;" alt="{\displaystyle ~\exp(-x)\!~\ln \!\left(1+{\frac {1}{x}}\right)~}"></span>
</p>
(red).
<a href="//commons.wikimedia.org/wiki/User:Domitori" title="User:Domitori">Domitori</a>
<span class="int-own-work" lang="en">Own work</span>