Fractional Derivative of Sine

α

0.5

2D view

3D view

This Demonstration calculates fractional derivatives of the sine function.

The definition of the fractional derivative of a function

y(x)

is

-α

d

y(x)

d

-α

x

=

1

Γ(a)

x

∫

0

α-1

(x-t)

y(t)dt

,

for

α>0

and

x>0

, and

α

d

y(x)

d

α

x

=

m

d

m

x

m-α

d

y(x)

d

m-α

x

,

where

m

is any postive integer greater than

α

.

The fractional derivative of the sine function works out to be

α

d

α

x

sin(x)=sinx+

π

2

α

.

"3D View" plots the surface of the function

z=sinx+

π

2

α

and the red curve in either view is the graph of

sinx+

π

2

α

for constant α.