I know that the Laplace transform of a function f(t) on [ 0 , ∞ ) is defined as f ∗ ( θ ) = ∫ 0 ∞ e − θ t f ( t ) d t .But if I only have f ∗ ( θ ) = 3 + 10 θ 24 θ 2 + 22 θ + 3 , how to evaluate f(t)?

Name: I know that the Laplace transform of a function f(t) on [ 0 , ∞ ) is defined as f ∗ ( θ ) = ∫ 0 ∞ e − θ t f ( t ) d t .But if I only have f ∗ ( θ ) = 3 + 10 θ 24 θ 2 + 22 θ + 3 , how to evaluate f(t)?
Uploaded: 2022-02-23T17:22:57+00:00
Description: I know that the Laplace transform of a function f(t) on [ 0 , ∞ ) is defined as f ∗ ( θ ) = ∫ 0 ∞ e − θ t f ( t ) d t .But if I only have f ∗ ( θ ) = 3 + 10 θ 24 θ 2 + 22 θ + 3 , how to evaluate f(t)?

Question

I know that the Laplace transform of a function f(t) on   [  0  ,  ∞  ) is defined as      f    ∗    (  θ  )  =      ∫    0    ∞        e          −      θ      t        f  (  t  )        d    t  .But if I only have       f    ∗    (  θ  )  =                    3        +        10        θ                    24                  θ          2                +        22        θ        +        3            , how to evaluate f(t)?

Peyton Atkins · Accepted Answer

Use partial fractions to write      f    ∗        θ    =      4          7      (      6      θ      +      1      )        +      9          7      (      4      θ      +      3      )      And now apply the fact that if   g  (  t  )  =      e          −      a      t      , then      g    ∗    (  θ  )  =      1          θ      +      a      To find  f  (  t  )  =      1    84        e          −      3      t              /            4            (    8          e              7        t                  /                12              +    27    )

How to evaluate the following f^(**)(theta)=int_0^(oo) e^(-theta t) f(t)dt.

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