lim s → 0 + ∫ 0 ∞ a ( t ) e − s t d t ∫ 0 ∞ a ( t ) e − s t d t = f ( s )What is the meaning of the limit of this integral as s → 0 + .

Name: lim s → 0 + ∫ 0 ∞ a ( t ) e − s t d t ∫ 0 ∞ a ( t ) e − s t d t = f ( s )What is the meaning of the limit of this integral as s → 0 + .
Uploaded: 2022-02-23T17:22:57+00:00
Description: lim s → 0 + ∫ 0 ∞ a ( t ) e − s t d t ∫ 0 ∞ a ( t ) e − s t d t = f ( s )What is the meaning of the limit of this integral as s → 0 + .

Question

lim          s      →              0        +                  ∫    0    ∞    a  (  t  )      e          −      s      t        d  t      ∫    0    ∞    a  (  t  )      e          −      s      t        d  t  =  f  (  s  )What is the meaning of the limit of this integral as   s  →      0    +    .

Samsonitew7b · Accepted Answer

I am guessing that you would like to know       lim          s      →      0      +        f  (  s  )  =      lim          s      →      0      +            ∫    0    ∞    a  (  t  )      e          −      s      t        d  t. Let   (      s    n    ) be any sequence of positive numbers converging to 0 and set       f    n    (  t  )  =  a  (  t  )      e          −              s        n            t        →  a  (  t  ) as   n  →  ∞. Moreover,       |        f    n    (  t  )      |    ≤      |    a  (  t  )      |   so if   a  (  t  ) is Lebesgue integrable, then the limit equals       ∫    0    ∞    a  (  t  )  d  t by Lebesgue&#039;s Dominated convergence theorem. On the other hand, if a(t) is not Lebesgue integrable, then the limit need not exist. Consider   a  (  t  )  =                    |            sin      ⁡      (      t      )              |              t  , for example

lim_(s -> 0^+) int_0^(oo) a(t)e^(-st) dt. What is the meaning of the limit of this integral as s -> 0^+.

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