Why is lim s → 0 ∫ 0 ∞ sin ⁡ ( t ) t ⋅ e − s t d t = ∫ 0 ∞ sin ⁡ ( t ) t d t legitimate?

Question

Why is       lim          s      →      0            ∫          0              ∞                  sin      ⁡      (      t      )        t    ⋅      e          −      s      t        d  t  =      ∫          0              ∞                  sin      ⁡      (      t      )        t    d  t legitimate?

Christina Ward · Accepted Answer

The integral  F  (  s  )  =      ∫    0    ∞    f  (  t  ,  s  )    d  t  =      ∫    0    ∞              sin      ⁡      t        t        e          −      s      t          d  tconverges uniformly for   s  ⩾  0. This follows from the Dirichlet test since       |          ∫      0      x        sin    ⁡    t        d    t    |   is uniformly bounded and       e          −      s      t            /    t is decreasing and uniformly convergent to 0 as   t  →  ∞.Given uniform convergence to F(s) and continuity of   (  t  ,  s  )  ↦  f  (  t  ,  s  ) we can apply a basic theorem found in standard real analysis textbooks (Rudin, Apostol, Bartle, etc.) proving F to be continuous, so      lim          s      →      0        F  (  s  )  =  F  (  0  )  =      ∫    0    ∞              sin      ⁡      t        t      d  t

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