Modular Forms course lecture notes 9

Modular Forms course lecture notes 9 - (April 7 2011...

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( April 7, 2011 ) Modular forms and number theory exercises 16 Paul Garrett [email protected] http: / / e garrett/ [mfms 16.1] Some calculus: derive the ordinary differential equation Δ f ( | x | ) = f 00 ( | x | ) + n - 1 | x | f 0 ( | x | ) [mfms 16.2] More calculus: eliminate the first-derivative term in a differential equation f 00 + pf 0 + qf = 0 by letting f = u · w , expanding, setting the u 0 coefficient to 0, and solving the resulting differential equation for w . Do this for the differential equation above. [mfms 16.3]
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Unformatted text preview: Execute the previous normalization procedure, and then irregular singular point discussion, to see that the leading term in the asymptotics for solutions of f 00 ( | x | ) + n-1 | x | f ( | x | )-λf = 0 really is f ( | x | ) ∼ e ± √ λ ·| x | | x | n-2 2 [mfms 16.4] * Carry out the same calculus exercises for f ( x ) = P ( x ) u ( | x | ), where P is a homogeneous, harmonic, degree d polynomial. 1...
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