Fourier Transformation of Heviside Function

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Given that  

show that

Hence, deduce that

Further, deduce that

Solution:

                         
                                                                        
                                   
                                   

The series expansion of

From duplication formula, we have

Therefore, we have

Substituting for  cos(st) in Equation (1) ,we have

                            
 
 
         

                                 
                                         
       

                                
          

From the definition of Bessel function of first kind, we have

Rewriting the result, we have

                                                   

                                  
          

Therefore,

   
                                     

                                     

Applying the inverse Fourier transform, we have

                               

Interchanging s and t, we get

                                

                               
             














 

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