Let T ( t ) = 1 − ∫ τ = 0 ∞ a ( τ ) e − t / τ d τ .where t ∈ ( 0 , ∞ ). I know T(t) and would like to find a ( τ ). This looks like a Laplace transform but the limits of integration are the wrong way round. Can I find a ( τ ) using an integral transform?

Name: Let T ( t ) = 1 − ∫ τ = 0 ∞ a ( τ ) e − t / τ d τ .where t ∈ ( 0 , ∞ ). I know T(t) and would like to find a ( τ ). This looks like a Laplace transform but the limits of integration are the wrong way round. Can I find a ( τ ) using an integral transform?
Uploaded: 2022-02-23T17:22:57+00:00
Description: Let T ( t ) = 1 − ∫ τ = 0 ∞ a ( τ ) e − t / τ d τ .where t ∈ ( 0 , ∞ ). I know T(t) and would like to find a ( τ ). This looks like a Laplace transform but the limits of integration are the wrong way round. Can I find a ( τ ) using an integral transform?

Question

Let  T  (  t  )  =  1  −      ∫          τ      =      0        ∞    a  (  τ  )            e              −      t              /            τ        d  τ  .where   t  ∈  (  0  ,  ∞  ). I know T(t) and would like to find   a  (  τ  ). This looks like a Laplace transform but the limits of integration are the wrong way round. Can I find   a  (  τ  ) using an integral transform?

Miya Swanson · Accepted Answer

T  (  t  )  =  1  −      ∫    0    ∞    a  (  τ  )      e          −              t        τ                 d  τ      ∫    ∞    0    a      (                  1        τ              )        e          −      t      τ           d      (                  1        τ              )    =  1  −  T  (  t  )      ∫    0    ∞              1              τ        2              a      (                  1        τ              )        e          −      t      τ           d  τ  =  1  −  T  (  t  )            L              τ      →      t            {                  1                  τ          2                      a          (                        1          τ                    )        }    =  1  −  T  (  t  )            1              τ        2              a      (                  1        τ              )    =  δ  (  τ  )  −            L              t      →      τ              −      1            {    T    (    t    )    }    a      (                  1        τ              )    =      τ    2    δ  (  τ  )  −      τ    2              L              t      →      τ              −      1            {    T    (    t    )    }    a  (  τ  )  =            1              τ        2              δ      (                  1        τ              )    −            1              τ        2                        L              t      →              1        τ                    −      1            {    T    (    t    )    }

Can I invert this expression with an integral transform? T(t)=1-int_(r=0)^(oo) a(tau)e^(-t/tau)d tau

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