^{1}

^{*}

^{2}

^{*}

^{3}

^{*}

The classical example of no-where differentiable but everywhere continuous function is Weierstrass function. In this paper we have defined fractional order Weierstrass function in terms of Jumarie fractional trigonometric functions. The H?lder exponent and Box dimension of this new function have been evaluated here. It has been established that the values of H?lder exponent and Box dimension of this fractional order Weierstrass function are the same as in the original Weierstrass function. This new development in generalizing the classical Weierstrass function by use of fractional trigonometric function analysis and fractional derivative of fractional Weierstrass function by Jumarie fractional derivative, establishes that roughness indices are invariant to this generalization.

The concepts of fractional geometry, fractional dimensions are important branches of science to study the irregularity of a function, graph or signals [

a) Fractional Order Derivative of Jumarie Type

Jumarie [

In the above definition, the first expression is just the Riemann-Liouvelli fractional integration; the second line is Riemann-Liouvelli fractional derivative of order

b) Mittag-Leffler Function and Its Jumarie Type Fractional Derivative: One and Two Parameter Type

1) One Parameter Mittag-Leffler Function

The Mittag-Leffler function [

This function plays a crucial role in classical calculus for

We now consider the Mittag-Leffler function in the following form in infinite series representation for

Then taking Jumarie fractional derivative of order

Like the exponential function;

Jumarie in [

Differentiating both side with respect to x and y of

First consider y a constant, and we fractionally differentiate w.r.t. x by Jumarie derivative

Now we consider x as constant and do the following steps

Here we put equivalence of

Jumarie fractional derivative of constant is zero. Therefore the RHS of above two expressions are equal, from that we get the following

The above two may be equated to a constant say

Using definition

Comparing real and imaginary part in above derived relation we get the following

This is very useful relation as in conjugation with classical trigonometric functions, and we will be using these relations in our analysis of fractional Weierstrass function and its fractional derivative.

2) Two Parameter Mittag-Leffler Function

The other important function is the two parameter Mittag-Leffler function denoted by

The functions (2) and (6) play important role in fractional calculus, also we note that

Again we derive Jumarie derivative of order

where

c) Jumarie Definition of Fractional Sine and Cosine Function and Their Fractional Derivative: Both One Parameter and Two Parameter Type

1) One Parameter Sine and Cosine Function

Jumarie [

From

The series representation of

Taking term by term Jumarie derivative we get,

The series presentation of

Taking term by term Jumarie derivative we get

Thus we get

2) Two Parameter Sine and Cosine Function

Let us define the two parameter sine and cosine functions

Now with this and with definition of two parameter Mittag-Leffler function (3) with imaginary argument we get the following useful identity

Now for

Thus we get a very useful relation

Similarly it can be shown that

Now we calculate the Jumarie type fractional order derivative of

On the other hand the Jumarie type fractional order derivative of

We obtain

Similarly the Jumarie type fractional order derivative of

a) Lipschitz Hölder Exponent (LHE)

A function is said to have LHE [

where

unless

Consider the function:

b) Holder Continuity

A continuous function

where

c) Fractional Dimension

Fractional dimension (d) or box dimension [

Again if H be the Hurst exponent then the relation between the above Holder exponents are

In 1872 K. Weierstrass [

where b is odd-integer. He proved that this function is continuous for all

In reference [

We define the fractional Weierstrass Function in terms of Jumarie [

where,

We only are stating some lemmas which will be used to characterize the fractional Weierstrass function and its fractional derivative.

Lemma 1:

Let f be function continuous in interval

Suppose

1)

then the dimension [

2) Suppose

Theorem 1: The Holder exponent of fractional Weierstrass function

Proof: We calculate

From the series expansion of

Choose

tion that is

With

where the constant

sion it is clear that fractional Weierstrass function is also Holder continuous with Holder exponent

Many authors found the fractional derivative of the continuous but nowhere differentiable function that is Weierstrass Function [

We used in above derivation the identity

Since if

gent if

Again if

is a divergent series for

This shows that

Theorem 2:

exists when

Theorem 3: The Holder exponent of

Proof: Let

denotes

From the series expansion of

Choose

With

where

with Holder exponent

The fractional Weierstrass function is a continuous function for all real values of the arguments, and its box dimension and Holder exponent are independent of fractional order that incorporates to the fractional Weierstrass functions. Again the Box dimension of fractional derivative of the fractional Weierstrass increases with increase of order of fractional derivative. This invariant nature of the roughness index of fractional Weierstrass function when generalized with fractional trigonometric function is remarkable. The other embodiment in similar lines as in this paper to get different fractional Weierstrass function is under development.

Acknowledgments are to Board of Research in Nuclear Science (BRNS), Department of Atomic Energy Government of India for financial assistance received through BRNS research project no. 37(3)/14/46/2014-BRNS with BSC BRNS, title “Characterization of unreachable (Holderian) functions via Local Fractional Derivative and Deviation Function”. Authors are also thankful to the reviewer for his valuable comments which has helped to improve the paper.

UttamGhosh,SusmitaSarkar,ShantanuDas, (2015) Fractional Weierstrass Function by Application of Jumarie Fractional Trigonometric Functions and Its Analysis. Advances in Pure Mathematics,05,717-732. doi: 10.4236/apm.2015.512065