UNIVERSITY OF SOUTH AFRICA (UNISA)
College of Science, Engineering and Technology
⋄
ASSIGNMENT 03
Question 4 — Year Module 2026
⋄
Module Code: MAT4847
Module Name: Partial Differential Equations I
Assignment No.: Assignment 03
Due Date: 05 October 2026
Semester: Year Module 2026
Submitted in partial fulfilment of the requirements for
Partial Differential Equations I (MAT4847)
at the University of South Africa.
, UNISA | MAT4847 Assignment 03 – Question 4
Question 4(a): Fourier Transform of the Antiderivative of p
Question. Assume that p is a piecewise continuous absolutely integrable function on R, with
F(p)(0) = 0. If
Z x
q(x) = p(α) dα, x ∈ (−∞, ∞),
−∞
show that (explaining clearly all steps)
F(p)(α)
F(q)(α) = , α ̸= 0.
iα
4(a).1 Step 1: Differentiate q and identify the relationship with p
By the Fundamental Theorem of Calculus, differentiating
Z x
q(x) = p(α) dα
−∞
with respect to x gives
q ′ (x) = p(x).
Hence q is an antiderivative of p.
4(a).2 Step 2: Recall the Fourier transform and its derivative property
The Fourier transform of a function f is defined as
Z ∞
F(f )(α) = f (x) e−iαx dx.
−∞
A standard property of the Fourier transform states that if f is absolutely integrable and f ′
exists, then
F(f ′ )(α) = iα F(f )(α),
provided f (x) → 0 as x → ±∞.
We verify this property for q directly by integration by parts in the next step.
Page 1 of 8
College of Science, Engineering and Technology
⋄
ASSIGNMENT 03
Question 4 — Year Module 2026
⋄
Module Code: MAT4847
Module Name: Partial Differential Equations I
Assignment No.: Assignment 03
Due Date: 05 October 2026
Semester: Year Module 2026
Submitted in partial fulfilment of the requirements for
Partial Differential Equations I (MAT4847)
at the University of South Africa.
, UNISA | MAT4847 Assignment 03 – Question 4
Question 4(a): Fourier Transform of the Antiderivative of p
Question. Assume that p is a piecewise continuous absolutely integrable function on R, with
F(p)(0) = 0. If
Z x
q(x) = p(α) dα, x ∈ (−∞, ∞),
−∞
show that (explaining clearly all steps)
F(p)(α)
F(q)(α) = , α ̸= 0.
iα
4(a).1 Step 1: Differentiate q and identify the relationship with p
By the Fundamental Theorem of Calculus, differentiating
Z x
q(x) = p(α) dα
−∞
with respect to x gives
q ′ (x) = p(x).
Hence q is an antiderivative of p.
4(a).2 Step 2: Recall the Fourier transform and its derivative property
The Fourier transform of a function f is defined as
Z ∞
F(f )(α) = f (x) e−iαx dx.
−∞
A standard property of the Fourier transform states that if f is absolutely integrable and f ′
exists, then
F(f ′ )(α) = iα F(f )(α),
provided f (x) → 0 as x → ±∞.
We verify this property for q directly by integration by parts in the next step.
Page 1 of 8
