Semester 5 / Calculus / Complex Functions / Integration over a Real Interval

Integration over a Real Interval

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Let f(t)=u(t)+iv(t)f(t) = u(t) + iv(t) for a≤t≤ba \le t \le b, where uu and vv are real-valued continuous functions of the real variable tt.

∫abf(t) dt=∫abu(t) dt+i∫abv(t) dt\int_a^b f(t)\,\text{d}t = \int_a^b u(t)\,\text{d}t + i\int_a^b v(t)\,\text{d}t

If U′(t)=u(t)U'(t) = u(t) and V′(t)=v(t)V'(t) = v(t) for each a≤t≤ba \le t \le b,

∫abf(t) dt=(U(b)−U(a))+i(V(b)−V(a))\int_a^b f(t)\,\text{d}t = (U(b) - U(a)) + i(V(b) - V(a))

Properties

Let f(t)=u(t)+iv(t)f(t) = u(t) + iv(t) and g(t)=p(t)+iq(t)g(t) = p(t) + iq(t) be continuous on [a,b][a,b].

Sum

∫ab(f(t)+g(t)) dt=∫abf(t) dt+∫abg(t) dt\int_a^b (f(t) + g(t))\,\text{d}t = \int_a^b f(t)\,\text{d}t + \int_a^b g(t)\,\text{d}t

Additivity over Subintervals

∫abf(t) dt=∫acf(t) dt+∫cbf(t) dt,a≤c≤b\int_a^b f(t)\,\text{d}t = \int_a^c f(t)\,\text{d}t + \int_c^b f(t)\,\text{d}t, \qquad a \le c \le b

Homogeneity

Let α=r+is\alpha = r + is be a complex constant.

∫abαf(t) dt=α∫abf(t) dt\int_a^b \alpha f(t)\,\text{d}t = \alpha \int_a^b f(t)\,\text{d}t

Reversal of Limits

∫abf(t) dt=−∫baf(t) dt\int_a^b f(t)\,\text{d}t = -\int_b^a f(t)\,\text{d}t

Fundamental Theorem

Let ff be continuously differentiable on [a,b][a,b].

∫abf′(t) dt=f(b)−f(a)\int_a^b f'(t)\,\text{d}t = f(b) - f(a)

Product

∫abf(t)g(t) dt=∫ab(up−vq) dt+i∫ab(uq+vp) dt\int_a^b f(t)g(t)\,\text{d}t = \int_a^b (u p - v q)\,\text{d}t + i\int_a^b (u q + v p)\,\text{d}t
Written by September 13, 2026 3 min read
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