Expression 8: "d" Subscript, 2 , Baseline left parenthesis, "x" , right parenthesis equals StartFraction, 1 Over 60 , EndFraction StartFraction, 4 Superscript, "s" , Baseline gamma left parenthesis, 1 half plus "s" , right parenthesis Over pi Superscript, 1 half , Baseline StartAbsoluteValue, gamma left parenthesis, negative "s" , right parenthesis , EndAbsoluteValue , EndFraction Start integral from 0 to 20, end integral, StartFraction, "f" left parenthesis, "x" , right parenthesis minus "f" left parenthesis, "x" minus "y" , right parenthesis Over abs left parenthesis, "y" , right parenthesis Superscript, 1 plus 2 "s" , Baseline plus 10 Superscript, negative 5 , Baseline , EndFraction "d" "y"d2x=1604sγ12+sπ12γ−s∫200fx−fx−yabsy1+2s+10−5dy
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fractional derivative
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Expression 10: "d" Subscript, 1 , Baseline left parenthesis, "x" , right parenthesis equals StartFraction, 1 Over gamma left parenthesis, 1 minus "s" , right parenthesis , EndFraction StartFraction, "d" Over "d" "x" , EndFraction Start integral from 0 to "x" , end integral, StartFraction, "f" left parenthesis, "t" , right parenthesis Over left parenthesis, "x" minus "t" , right parenthesis Superscript, "s" , Baseline plus 10 to the negative 5th power , EndFraction "d" "t"d1x=1γ1−sddx∫x0ftx−ts+10−5dt
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fractional integral
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Expression 12: "I" Subscript, 1 , Baseline left parenthesis, "x" , right parenthesis equals StartFraction, 1 Over gamma left parenthesis, "s" , right parenthesis , EndFraction Start integral from 0 to "x" , end integral, StartFraction, "f" left parenthesis, "t" , right parenthesis Over left parenthesis, "x" minus "t" , right parenthesis Superscript, 1 minus "s" , Baseline plus 10 to the negative 5th power , EndFraction "d" "t"I1x=1γs∫x0ftx−t1−s+10−5dt
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Expression 13: gamma left parenthesis, "x" , right parenthesis equals left parenthesis, "x" minus 1 , right parenthesis !γx=x−1!