The Fourier Integral Representation of a Function F Is Given $\int _ { - \infty } ^ { \infty } f ( x ) \cos ( \alpha x ) d x$

question 24

Multiple Choice

The Fourier integral representation of a function f is given by

• A) \(\int _ { - \infty } ^ { \infty } f ( x ) \cos ( \alpha x ) d x\)
• B) \(\int _ { - \infty } ^ { \infty } f ( x ) \sin ( \alpha x ) d x\)
• C) \(\int _ { 0 } ^ { \infty } \left[ \int _ { - \infty } ^ { \infty } f ( t ) \cos ( \alpha t ) d t \cos ( \alpha x ) + \int _ { - \infty } ^ { \infty } f ( t ) \sin ( \alpha t ) d t \sin ( \alpha x ) \right] d \alpha\)
• D) \(\int _ { 0 } ^ { \infty } \left[ \int _ { - \infty } ^ { \infty } f ( t ) \cos ( \alpha t ) d t \cos ( \alpha x ) + \int _ { - \infty } ^ { \infty } f ( t ) \sin ( \alpha t ) d t \sin ( \alpha x ) \right] d \alpha / \pi\)
• E) \(\int _ { 0 } ^ { \infty } \left[ \int _ { - \infty } ^ { \infty } f ( t ) \cos ( \alpha t ) d t \cos ( \alpha x ) + \int _ { - \infty } ^ { \infty } f ( t ) \sin ( \alpha t ) d t \sin ( \alpha x ) \right] d \alpha / ( 2 \pi )\)

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