this case, the expansion in (B13) is not valid, as can be seen by comparing (B13) with the results in figure 10. 

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[2] L. D. Landau, J. Phys. (U.S.S.R.) 5 , 71 (1941). 

[3] A. Griffin, T. Nikuni and E. Zaremba, Bose-condensed gases at finite temperatures , (Cam- bridge, N.Y., 2009). 

[4] R. Meppelink, S. B. Koller, and P. van der Straten, Phys. Rev. A 80 , 043605 (2009). 

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[13] See, for example, T. J. Greytak in Low Temperature Physics LT13 , ed. by K. D. Timmermans, W. J. O’Sullivan, and E. F. Hammel (Plenum Press, N.Y. 1974), page 9. 

[14] W. F. Vinen in Physics of Quantum Fluids , ed. by R. Kubo and F. Takano (Syokabo Publishing Company, Tokyo, 1970), page 1. 

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[19] A. Griffin, Excitations in a Bose-condensed liquid , (Cambridge, N.Y. 1993), chapter 2. 

[20] D. M. Stamper-Kurn, A. P. Chikkatur, A. G¨ orlitz, S. Inouye, S. Gupta, D. E. Pritchard, and 

