| Summary: | In the most common literature about fractional calculus, we find that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>D</mi><mrow><mi>t</mi></mrow><mi>α</mi><mprescripts></mprescripts><mi>a</mi><none></none></mmultiscripts><mi>f</mi><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>=</mo><mmultiscripts><mi>I</mi><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mi>α</mi></mrow><mprescripts></mprescripts><mi>a</mi><none></none></mmultiscripts><mi>f</mi><mfenced open="(" close=")"><mi>t</mi></mfenced></mrow></semantics></math></inline-formula> is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the definitions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>I</mi><mrow><mi>t</mi></mrow><mi>α</mi><mprescripts></mprescripts><mi>a</mi><none></none></mmultiscripts><mi>f</mi><mfenced open="(" close=")"><mi>t</mi></mfenced></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>D</mi><mrow><mi>t</mi></mrow><mi>α</mi><mprescripts></mprescripts><mi>a</mi><none></none></mmultiscripts><mi>f</mi><mfenced open="(" close=")"><mi>t</mi></mfenced></mrow></semantics></math></inline-formula>. In this sense, we prove that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>D</mi><mrow><mi>t</mi></mrow><none></none><mprescripts></mprescripts><mn>0</mn><none></none></mmultiscripts><mi>f</mi><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>=</mo><mmultiscripts><mi>I</mi><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mi>α</mi></mrow><mprescripts></mprescripts><mn>0</mn><none></none></mmultiscripts><mi>f</mi><mfenced open="(" close=")"><mi>t</mi></mfenced></mrow></semantics></math></inline-formula> is true for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>=</mo><msup><mi>t</mi><mrow><mi>ν</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>log</mi><mi>t</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>=</mo><msup><mi>e</mi><mrow><mi>λ</mi><mi>t</mi></mrow></msup></mrow></semantics></math></inline-formula>, despite the fact that these derivations are highly non-trivial. Moreover, the corresponding formulas for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>D</mi><mrow><mi>t</mi></mrow><mi>α</mi><mprescripts></mprescripts><mrow><mo>−</mo><mo>∞</mo></mrow><none></none></mmultiscripts><msup><mfenced open="|" close="|"><mi>t</mi></mfenced><mrow><mo>−</mo><mi>δ</mi></mrow></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>I</mi><mrow><mi>t</mi></mrow><mi>α</mi><mprescripts></mprescripts><mrow><mo>−</mo><mo>∞</mo></mrow><none></none></mmultiscripts><msup><mfenced open="|" close="|"><mi>t</mi></mfenced><mrow><mo>−</mo><mi>δ</mi></mrow></msup></mrow></semantics></math></inline-formula> found in the literature are incorrect; thus, we derive the correct ones, proving in turn that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>D</mi><mrow><mi>t</mi></mrow><mi>α</mi><mprescripts></mprescripts><mrow><mo>−</mo><mo>∞</mo></mrow><none></none></mmultiscripts><msup><mfenced open="|" close="|"><mi>t</mi></mfenced><mrow><mo>−</mo><mi>δ</mi></mrow></msup><mo>=</mo><mmultiscripts><mi>I</mi><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mi>α</mi></mrow><mprescripts></mprescripts><mrow><mo>−</mo><mo>∞</mo></mrow><none></none></mmultiscripts><msup><mfenced open="|" close="|"><mi>t</mi></mfenced><mrow><mo>−</mo><mi>δ</mi></mrow></msup></mrow></semantics></math></inline-formula> holds true.
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