I'm trying to get estimate on the following partial summation using Abel-Plana Summation formula: ∑n=1xsin2⁡(n)n" role="presentation" style="margin: 0px; padding: 0px; border: 0px; font-variant:...

I'm trying to get estimate on the following partial summation using Abel-Plana Summation formula:

∑n=1xsin2⁡(n)n" role="presentation" style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; vertical-align: baseline; box-sizing: inherit; display: inline; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">∑n=1xsin2(n)n∑n=1xsin2⁡(n)n

I can handle the first integral in the formula but I'm stuck at the following functional:

T(x)=∫0∞(sin2⁡(x+iy)(x+iy)−sin2⁡(x−iy)(x−iy))(e2πy−1)dy" role="presentation" style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; vertical-align: baseline; box-sizing: inherit; display: inline; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">T(x)=∫∞0(sin2(x+iy)(x+iy)−sin2(x−iy)(x−iy))(e2πy−1)dyT(x)=∫0∞(sin2⁡(x+iy)(x+iy)−sin2⁡(x−iy)(x−iy))(e2πy−1)dy

Questions:

(1) Sharp upper and lower bounds onT(x)" role="presentation" style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; vertical-align: baseline; box-sizing: inherit; display: inline; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">T(x)T(x).

(2) Graph ofT(x)" role="presentation" style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; vertical-align: baseline; box-sizing: inherit; display: inline; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">T(x)T(x)(need an idea about it's growth).

Note: I tried to expandsin" role="presentation" style="margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; vertical-align: baseline; box-sizing: inherit; display: inline; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">sinsinas complex variable in terms of hyperbolic functions ; but then I can't handle integral after expansion .