That’s fair. What I was trying to convey (in my notation, deliberately annoying to reduce a sense of familiarity) is limx→0−∑∞k=1kekxcos(kx)=−112 . Any ideas for how I could write that better?
I added some actual values for concreteness. Hopefully that helps.
Do we have the value of the sum as a function of x, before going to the limit as x goes to 0 ? If yes, it would help (bonus points if it can be proven in a few lines).
That’s fair. What I was trying to convey (in my notation, deliberately annoying to reduce a sense of familiarity) is limx→0−∑∞k=1kekxcos(kx)=−112 . Any ideas for how I could write that better?
I added some actual values for concreteness. Hopefully that helps.
Do we have the value of the sum as a function of x, before going to the limit as x goes to 0 ? If yes, it would help (bonus points if it can be proven in a few lines).
Mathematica yields e(1+i)x(e2ix−4e(1+i)x+e2x+e(2+2i)x+1)2(−ex+eix)2(−1+e(1+i)x)2. That probably simplifies though.