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Prof. David R. Jackson ECE Dept. Spring 2014 Notes 30 ECE 6341 1

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2-D Stationary Phase Method 2D stationary phase point: Assume 2

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2-D Stationary Phase (cont.) Denote Then 3

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2-D Stationary Phase (cont.) Let where and We then have 4

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2-D Stationary Phase (cont.) Let We then have 5

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2-D Stationary Phase (cont.) Complete the square: 6

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2-D Stationary Phase (cont.) Now use The integral I is then so 7

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2-D Stationary Phase (cont.) The integral I is then in the form of the product of two integrals: 8

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2-D Stationary Phase (cont.) Integral in s : This has the form Use Recall that 9

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2-D Stationary Phase (cont.) Define: Integral in t : 10

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2-D Stationary Phase (cont.) Hence Then we have Use 11

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2-D Stationary Phase (cont.) We then have 12

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2-D Stationary Phase (cont.) or 13

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2-D Stationary Phase (cont.) Important special case: In this case: and so 14

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2-D Stationary Phase (cont.) where We then have: 15

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2-D Stationary Phase (cont.) where Hence, the final result is 16

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