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24. Let L be a partial differential operator of the form 2² 2² 2² L = a +b- +c- მე2 მყ2 ахду O with a, b, c constants. Assume that L commutes with rotations in the sense that whenever ƒ is a C² function on C, then (Lƒ) ○ pe = L(ƒ ope) for any rotation pe(z) = e¹ºz. Prove that £ must be a constant multiple of the Laplacian.

24. Let L be a partial differential operator of the form 2² 2² 2² L = a +b- +c- მე2 მყ2 ахду O with a, b, c constants. Assume that L commutes with rotations in the sense that whenever ƒ is a C² function on C, then (Lƒ) ○ pe = L(ƒ ope) for any rotation pe(z) = e¹ºz. Prove that £ must be a constant multiple of the Laplacian.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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24. Let L be a partial differential operator of the form 2² 2² 2² L = a +b- +c- dx² дуг əxəy' with a, b, c constants. Assume that L commutes with rotations in the sense that whenever f is a C² function on C, then (Lf) o pe = L(fope) for any rotation pe(z) = ez. Prove that L must be a constant multiple of the Laplacian. Graduate complex analysis - Greene and Krautz
Transcribed Image Text:24. Let L be a partial differential operator of the form 2² 2² 2² L = a +b- +c- dx² дуг əxəy' with a, b, c constants. Assume that L commutes with rotations in the sense that whenever f is a C² function on C, then (Lf) o pe = L(fope) for any rotation pe(z) = ez. Prove that L must be a constant multiple of the Laplacian. Graduate complex analysis - Greene and Krautz
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