| Summary: | Motivated from studies on anomalous relaxation and diffusion, we show that the memory function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> of complex materials, that their creep compliance follows a power law, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>∼</mo><msup><mi>t</mi><mi>q</mi></msup></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup></mrow></semantics></math></inline-formula>, is proportional to the fractional derivative of the Dirac delta function, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mrow><msup><mi mathvariant="normal">d</mi><mi>q</mi></msup><mi>δ</mi><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>0</mn><mo>)</mo></mrow></mrow><mrow><mi mathvariant="normal">d</mi><msup><mi>t</mi><mi>q</mi></msup></mrow></mfrac></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup></mrow></semantics></math></inline-formula>. This leads to the finding that the inverse Laplace transform of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>s</mi><mi>q</mi></msup></semantics></math></inline-formula> for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup></mrow></semantics></math></inline-formula> is the fractional derivative of the Dirac delta function, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mrow><msup><mi mathvariant="normal">d</mi><mi>q</mi></msup><mi>δ</mi><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>0</mn><mo>)</mo></mrow></mrow><mrow><mi mathvariant="normal">d</mi><msup><mi>t</mi><mi>q</mi></msup></mrow></mfrac></semantics></math></inline-formula>. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><msup><mi>s</mi><mi>q</mi></msup><mrow><msup><mi>s</mi><mi>α</mi></msup><mo>∓</mo><mi>λ</mi></mrow></mfrac></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo><</mo><mi>q</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup></mrow></semantics></math></inline-formula>, which is the fractional derivative of order <i>q</i> of the Rabotnov function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mo>±</mo><mi>λ</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>t</mi><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mi>E</mi><mrow><mi>α</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>α</mi></mrow></msub><mrow><mo>(</mo><mo>±</mo><mi>λ</mi><msup><mi>t</mi><mi>α</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The fractional derivative of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup></mrow></semantics></math></inline-formula> of the Rabotnov function, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mo>±</mo><mi>λ</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> produces singularities that are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of <i>q</i> in association with the recurrence formula of the two-parameter Mittag–Leffler function.
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